thinking.about.logic..classic.essays

THINKING ABOUT

LOGICClassic Essays

EDITED BY

STEVEN M. CAHNROBERT B. TALISSE

SCOTT F. AIKIN

A Member of the Perseus Books Groupwww.westviewpress.comwww.perseusacademic.com

“A very fi ne collection of essays on the nature of logic as it relates to broad philosophical concerns. This volume should serve as a valuable tool for students to see the applicability of formal logical techniques and results. It is a useful antidote to what can—and for students often does—seem to be the insulated nature of formal logic.” —David Boersema, Pacifi c University

“The formal logic that we teach in our introductory classes is simple, elegant, and asthis thoughtfully conceived collection reminds us, often extremely puzzling. Thinking about Logic will motivate students to refl ect both on fundamental notions of basic logic and on the relationship between formal reasoning and reasoning in other contexts. Undergraduate courses will be enriched and improved by adding it to the list of required readings.”

—Danielle Macbeth, Haverford College

“Anyone who studies formal logic is bound at some point to wonder what logic is all about. Thinking about the philosophical issues as you go along can help bring logic to life. This col-lection of classic papers is the ideal place to begin an exploration of the philosophy of logic.”

—Derek Turner, Connecticut College

Thinking about Logic is an accessible and thought-provoking collection of classic articles in the philosophy of logic. An ideal companion to any formal logic course or textbook, thisvolume illuminates how logic relates to perennial philosophical issues about knowledge, meaning, rationality, and reality. The editors have selected each essay for its brevity, clarity,and impact, and they have included insightful introductions and discussion questions. The puzzles raised will help readers acquire a more thorough understanding of fundamental logical concepts and a fi rmer command of the connections between logic and other areas of philo-sophical study: epistemology, philosophy of language, philosophy of science, and metaphysics.

STEVEN M. CAHN is professor of philosophy at The City University of New YorkGraduate Center.

ROBERT B. TALISSE is professor of philosophy at Vanderbilt University.

SCOT T F. AIKIN is a senior lecturer at Vanderbilt University.

COVER DESIGN: MIGUEL SANTANA & WENDY HALITZER

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THINKING ABOUT LOGIC

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THINKING ABOUT LOGIC

Classic Essays

EDITED BY

Steven M. CahnCity University of New York Graduate Center

Robert B. TalisseVanderbilt University

Scott F. AikinVanderbilt University

A MEMBER OF THE PERSEUS BOOKS GROUP

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Copyright © 2011 by Westview PressPublished by Westview Press,A Member of the Perseus Books Group

Every effort has been made to secure required permissions to use all

images, maps, and other art included in this volume.

The Credits on page 215 constitute an extension of this Copyright

Page.

All rights reserved. Printed in the United States of America. No part ofthis book may be reproduced in any manner whatsoever without writtenpermission except in the case of brief quotations embodied in criticalarticles and reviews. For information, address Westview Press, 2465Central Avenue, Boulder, CO 80301.Find us on the World Wide Web at www.westviewpress.com.

Westview Press books are available at special discounts for bulkpurchases in the United States by corporations, institutions, and otherorganizations. For more information, please contact the Special MarketsDepartment at the Perseus Books Group, 2300 Chestnut Street, Suite200, Philadelphia, PA 19103, or call (800) 810-4145, x5000, or [email protected].

Library of Congress Cataloging-in-Publication DataThinking about logic : classic essays / edited by Steven M. Cahn, RobertB. Talisse. Scott F. Aikin.

p. cm.ISBN 978-0-8133-4469-0 (alk. paper)1. Logic. I. Cahn, Steven M. II. Talisse, Robert B. III. Aikin, Scott F. BC6.T45 2011160—dc22

2010017146

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CONTENTS

Preface, vii

I . LOGIC AND KNOWLEDGE

1 What the Tortoise Said to AchillesLewis Carroll 3

2 What Achilles Said to the TortoiseW. J. Rees 9

3 What Achilles Should Have Said to the TortoiseJ. F. Thomson 19

Questions 35

II . LOGIC AND DEFINITION

4 The Runabout Inference-TicketA. N. Prior 39

5 Roundabout the Runabout Inference-TicketJ. T. Stevenson 43

6 Tonk, Plonk and PlinkNuel D. Belnap 51

Questions 59

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I I I . LOGIC AND INFERENCE

7 A Counterexample to Modus PonensVann McGee 63

8 Not a Counterexample to Modus PonensE. J. Lowe 79

9 Assumptions and the Supposed Counterexamples to Modus Ponens D. E. Over 85

Questions 93

IV. LOGIC AND FREEDOM

10 ‘It Was to Be’Gilbert Ryle 97

11 FatalismRichard Taylor 123

12 Time, Truth, and AbilityRichard Taylor and Steven M. Cahn 137

Questions 145

V. LOGIC AND REALITY

13 The Justification of DeductionSusan Haack 149

14 The Problem of Counterfactual ConditionalsNelson Goodman 163

15 On What There IsWillard V. Quine 189

Questions 211

About the Contributors, 213Source Credits, 215

vi Contents

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PREFACE

The study of logic raises philosophical questions about knowledge,meaning, rationality, and reality. Most textbooks in the field, however,pass by such matters in order to focus exclusively on principles of for-mal reasoning and strategies of formal proof. The assumption appearsto be that discussion of the larger philosophical themes would be adistraction from the mastery of technique. Yet one can find a numberof short, gripping, accessible essays that display the subject’s widersignificance.

This reader contains fifteen of these classic articles in the philosophyof logic. The collection is usable with any logic textbook and can bringadditional perspective to any logic course. The selections have beenchosen for their brevity, clarity, and impact. They deepen understandingof fundamental concepts of logic, while displaying connections be-tween logic and other areas of philosophy, including metaphysics, epis-temology, philosophy of science, and philosophy of language.

We wish to take this opportunity to thank our editor, Karl Yambert,for his firm support and wise counsel. We also appreciate the kind as-sistance of the staff at Westview Press.

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I

LOGIC

AND

KNOWLEDGE

If you deduce from two propositions, p and q, a third, r, then pre-sumably you first need to believe s, that p and q imply r. You thenalso need to believe t, that p, q, and s imply r. And so on. Thus itseems that any inference requires an infinite number of steps.

To reason correctly from premises to conclusion, we needrules of inference. Are they themselves further premises? If so,Lewis Carroll’s story of Achilles and the Tortoise demonstratesthe inherent regress that develops. W. J. Rees suggests that theapparent regress is a consequence of confusing premises withmeta-premises for inference. J. F. Thomson argues that agreeingto rules for inference is implicit in committing to any premise,and so the rules themselves should not be treated as additionalpremises.

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1

What the Tortoise Said to Achilles

Lewis Carroll

Achilles had overtaken the Tortoise, and had seated himself comfort-ably on its back.

“So you’ve got to the end of our race-course?” said the Tortoise.“Even though it does consist of an infinite series of distances? I thoughtsome wiseacre or other had proved that the thing couldn’t be done?”

“It can be done,” said Achilles. “It has been done! Solvitur ambu-lando. You see the distances were constantly diminishing; and so—”

“But if they had been constantly increasing?” the Tortoise inter-rupted. “How then?”

“Then I shouldn’t be here,” Achilles modestly replied; “and youwould have got several times round the world, by this time!”

“You flatter me—flatten, I mean,” said the Tortoise; “for you are aheavy weight, and no mistake! Well now, would you like to hear of arace-course, that most people fancy they can get to the end of in two orthree steps, while it really consists of an infinite number of distances,each one longer than the previous one?”

“Very much indeed!” said the Grecian warrior, as he drew from hishelmet (few Grecian warriors possessed pockets in those days) an enor-mous note-book and a pencil. “Proceed! And speak slowly, please!Shorthand isn’t invented yet!”

3


“That beautiful First Proposition of Euclid!” the Tortoise murmureddreamily. “You admire Euclid?”

“Passionately! So far, at least, as one can admire a treatise thatwo’n’t be published for some centuries to come!”

“Well, now, let’s take a little bit of the argument in that First Propo-sition—just two steps, and the conclusion drawn from them. Kindlyenter them in your note-book. And in order to refer to them conve-niently, let’s call them A, B, and Z:—

(A) Things that are equal to the same are equal to each other.(B) The two sides of this Triangle are things that are equal to the

same.(Z) The two sides of this Triangle are equal to each other.

Readers of Euclid will grant, I suppose, that Z follows logically fromA and B, so that any one who accepts A and B as true, must accept Z astrue?”

“Undoubtedly! The youngest child in a High School—as soon asHigh Schools are invented, which will not be till some two thousandyears later—will grant that.”

“And if some reader had not yet accepted A and B as true, he mightstill accept the sequence as a valid one, I suppose?”

“No doubt such a reader might exist. He might say ‘I accept as truethe Hypothetical Proposition that, if A and B be true, Z must be true;but, I don’t accept A and B as true.’ Such a reader would do wisely inabandoning Euclid, and taking to football.”

“And might there not also be some reader who would say ‘I acceptA and B as true, but I don’t accept the Hypothetical’?”

“Certainly there might. He, also, had better take to football.”“And neither of these readers,” the Tortoise continued, “is as yet

under any logical necessity to accept Z as true?”

4 LEWIS CARROLL


“Quite so,” Achilles assented.“Well, now, I want you to consider me as a reader of the second kind,

and to force me, logically, to accept Z as true.”“A tortoise playing football would be—” Achilles was beginning“—an anomaly, of course,” the Tortoise hastily interrupted. “Don’t

wander from the point. Let’s have Z first, and football afterwards!”“I’m to force you to accept Z, am I?” Achilles said musingly. “And

your present position is that you accept A and B, but you don’t acceptthe Hypothetical—”

“Let’s call it C,” said the Tortoise.“—but you don’t accept

(C) If A and B are true, Z must be true.”

“That is my present position,” said the Tortoise.“Then I must ask you to accept C.”“I’ll do so,” said the Tortoise, “as soon as you’ve entered it in that

note-book of yours. What else have you got in it?”“Only a few memoranda,” said Achilles, nervously fluttering the

leaves: “a few memoranda of—of the battles in which I have distin-guished myself!”

“Plenty of blank leaves, I see!” the Tortoise cheerily remarked. “Weshall need them all!” (Achilles shuddered.) “Now write as I dictate:—

(A) Things that are equal to the same are equal to each other.(B) The two sides of this Triangle are things that are equal to the same.(C) If A and B are true, Z must be true.(Z) The two sides of this Triangle are equal to each other.”

“You should call it D, not Z,” said Achilles. “It comes next to theother three. If you accept A and B and C, you must accept Z.”

1 What the Tortoise Said to Achilles 5


“And why must I?”“Because it follows logically from them. If A and B and C are true,

Z must be true. You don’t dispute that, I imagine?”“If A and B and C are true, Z must be true,” the Tortoise thoughtfully

repeated. “That’s another Hypothetical, isn’t it? And, if I failed to see itstruth, I might accept A and B and C, and still not accept Z, mightn’t I?”

“You might,” the candid hero admitted; “though such obtusenesswould certainly be phenomenal. Still, the event is possible. So I mustask you to grant one more Hypothetical.”

“Very good. I’m quite willing to grant it, as soon as you’ve writtenit down. We will call it

(D) If A and B and C are true, Z must be true.

Have you entered that in your note-book?”“I have!” Achilles joyfully exclaimed, as he ran the pencil into its

sheath. “And at last we’ve got to the end of this ideal race-course! Nowthat you accept A and B and C and D, of course you accept Z.”

“Do I?” said the Tortoise innocently. “Let’s make that quite clear. Iaccept A and B and C and D. Suppose I still refused to accept Z?”

“Then Logic would take you by the throat, and force you to do it!”Achilles triumphantly replied. “Logic would tell you ‘You ca’n’t helpyourself. Now that you’ve accepted A and B and C and D, you mustaccept Z!’ So you’ve no choice, you see.”

“Whatever Logic is good enough to tell me is worth writing down,”said the Tortoise. “So enter it in your book, please. We will call it

(E) If A and B and C and D are true, Z must be true.

Until I’ve granted that, of course I needn’t grant Z. So it’s quite anecessary step, you see?”

6 LEWIS CARROLL


“I see,” said Achilles; and there was a touch of sadness in his tone.Here the narrator, having pressing business at the Bank, was obliged

to leave the happy pair, and did not again pass the spot until somemonths afterwards. When he did so, Achilles was still seated on theback of the much-enduring Tortoise, and was writing in his note-book,which appeared to be nearly full. The Tortoise was saying “Have yougot that last step written down? Unless I’ve lost count, that makes athousand and one. There are several millions more to come. And wouldyou mind, as a personal favour, considering what a lot of instructionthis colloquy of ours will provide for the Logicians of the NineteenthCentury—would you mind adopting a pun that my cousin the Mock-Turtle will then make, and allowing yourself to be re-named Taught-Us?”

“As you please!” replied the weary warrior, in the hollow tones ofdespair, as he buried his face in his hands. “Provided that you, for yourpart, will adopt a pun the Mock-Turtle never made, and allow yourselfto be re-named A Kill-Ease!”

1 What the Tortoise Said to Achilles 7


2

What Achilles Said to the Tortoise

W. J. Rees

(Being a revised account of a famous interview, first reported in Mind in April, 1895, by Lewis Carroll.)

Achilles, as is well known, had overtaken the Tortoise, and had seatedhimself comfortably on its back, thus proving that he could overtakethe Tortoise, although the race-course consisted of an infinite series ofdistances.

Whereupon the Tortoise turned mathematician once more and pre-sented Achilles with a most perplexing problem in logic.

“Would you like to hear,” he said, “of a race-course that most peoplefancy they can get to the end of in two or three steps, while it reallyconsists of an infinite number of distances, each one longer than theprevious one?”

“Very much, indeed!” said the Grecian warrior, as he drew from hishelmet an enormous note-book and a pencil. “Proceed! And speakslowly, please! Short-hand isn’t invented yet!”

“That beautiful First Proposition of Euclid!” the Tortoise murmureddreamily. “You admire Euclid?”

“Passionately! So far, at least, as one can admire a treatise that won’tbe published for some centuries to come!”

9


“Well, now, let’s take a little bit of the argument in that First Propo-sition—just two steps, and the conclusion drawn from them. Kindlyenter them in your note-book. And in order to refer to them conve-niently, let’s call them A, B, and Z:

(A) Things that are equal to the same are equal to each other.(B) The two sides of this triangle are things that are equal to the

same.

Therefore

(Z) the two sides of this triangle are equal to each other.

Readers of Euclid will grant, I suppose, that Z follows logically fromA and B, so that anyone who accepts A and B as true, must accept Z astrue?”

“Undoubtedly! The youngest child in a High School—as soon asHigh Schools are invented, which will not be till some two thousandyears later—will grant that.”

“And if some reader had not yet accepted A and B as true, he mightstill accept the sequence as a valid one, I suppose?”

“No doubt such a reader might exist. He might say ‘I accept as truethe hypothetical proposition that, if A and B be true, Z is true; but, Idon’t accept A and B as true’. Such a reader would do wisely in aban-doning Euclid, and taking to football.”

“And might there not also be some reader who would say ‘I acceptA and B as true, but I don’t accept the hypothetical’?”

“Certainly there might. He, also, had better take to football.’’“And neither of these readers,” the Tortoise continued, “is as yet

under any logical necessity to accept Z as true?”

10 W. J. REES


“Quite so,” Achilles assented.“Well, now, I want you to consider me as a reader of the second kind,

and to present me with premisses which will enable me to deduce thetruth of Z.”

“You want me to present you with premisses which will enableyou to deduce the truth of Z?” Achilles said musingly. “And yourpresent position is that you accept A and B, but you don’t accept thehypothetical—”

“Let’s call it C,” said the Tortoise.“But you don’t accept

(C) If A and B are true, Z must be true.”

“That is my present position,” said the Tortoise.“Then I must ask you to accept C.”“I’ll do so,” said the Tortoise, “as soon as you’ve entered it in that

note-book of yours. Now write as I dictate:

(A) Things that are equal to the same are equal to each other.(B) The two sides of this triangle are things that are equal to the same.(C) If A and B are true, Z must be true.

Therefore

(Z) The two sides of this triangle are equal to each other.”

“Now I see what you are driving at,” said Achilles. “You want meto say that if one accepts A and B and C as true, then one must acceptZ as true?”

“Exactly! That’s what I wanted you to say.”

2 What Achilles Said to the Tortoise 11


“And then,” continued Achilles, “you are going to tell me that thisis another hypothetical—we’ll call it D—and that since you don’t ac-cept this hypothetical, you will still be able to accept A and B and C astrue, but not Z?”

“Precisely!”“And then, no doubt, you will ask me to enter this hypothetical again

among my other premisses, and say to you that, if you accept A and Band C and D as true, then you must accept Z as true?”

“I’ll grant you; that was roughly what I expected.”“Well,” said Achilles, “there’s certainly a future for this kind of job!

It will certainly go on until I shall have no more room in my note-bookto write down the premisses! Indeed, I should have to remain seatedon your back for all time, trying to find sufficient premisses to enableyou to deduce the truth of Z!”

“Very true!” said the Tortoise triumphantly. “Very true, indeed! Andnow do you see that you have discovered what no logician is going todiscover for centuries yet to come, an infinite regress?” (For the infiniteregress had not yet really been invented). “And do you see,” continuedthe Tortoise, “that no matter how long you try, you can never give mesufficient premisses to enable me to deduce the truth of Z?”

“You do flatter me!” said Achilles. “I’ll admit that I spotted theregress, but it was you who invented it.”

“I invented it!” said the tortoise, with great astonishment. “One doesnot need to invent it; it is just there, and one can’t do anything about it.”

“No,” said Achilles, “I’ll stick to my point. You yourself invented it.You invented it by compelling me to write down that condition C withthe other two conditions A and B, and so deceiving yourself into think-ing that C was on the same footing as A and B. Did you want to deceiveyourself, or were you just trying to tease me?”

“Neither, I assure you! But where else could you have inserted C?Can’t you see that if my accepting the hypothetical is a condition of

12 W. J. REES


my accepting Z as true, it must be put in with the conditions A andB?”

“Well, yes and no. It must be put in with A and B but not on the samefooting as A and B. By making C into an extra premiss you are substi-tuting, as Aristotle might some day put it, a hypothetical argument fora syllogistic argument. Instead of saying ‘A and B, therefore Z’, younow say ‘If A and B then Z, A and B, therefore Z’.”

“Why!” said the Tortoise, “even an intermediate student can see that.But this only brings in another hypothetical. And since I don’t acceptthis hypothetical, I can still accept all the premisses and deny the con-clusion. You still haven’t presented me with sufficient premisses to en-able me to deduce Z.”

“Not so fast!” replied Achilles, somewhat impatiently. “The troublewith you is that you don’t see that there are hypotheticals and hypo-theticals. You’ve never distinguished, have you, between first and sec-ond order hypotheticals?”

“When I did Honours,” said the Tortoise, “there was certainly nonews of them then.”

“Well, then, I must briefly tell you about them. If I may use wordswhich will only be understood in Oxford, and that only three thousandyears hence, I would say that a second order hypothetical is a hypo-thetical which has another hypothetical—this time, a first order hypo-thetical—as its apodosis.”

“Well, that is rather a mouthful! Could you not say it, please, in or-dinary standard English—which is not yet invented, of course, butwhich will be appreciated in Oxford three thousand years from now?”

“By all means!” said Achilles, somewhat surprised to find so muchhumility and foresight all at once.

“A second order hypothetical,” he continued, “states a conditionunder which another hypothetical is true. A hypothetical about a hy-pothetical! A two-storeyed hypothetical!”



“And, I suppose, this kind of hypothetical is important?”“Very important! Can’t you see that in a second order hypothetical

the condition is always a condition on which we can make a true con-ditional statement, and not, therefore, a condition which can itself beincluded in that conditional statement?”

“I see that most clearly,” said the Tortoise, “but what has this got todo with our argument?”

“This, surely: that the statement of the implication in a so-called hy-pothetical argument must generally take the form of a second order hy-pothetical, and not a first order hypothetical as would be the case witha syllogistic argument.”

“Could you give an example? We must have examples, you know!”“Well, let me see,” said Achilles, “instead of saying that, if one ac-

cepts A and B and C one must accept Z, we must now say that, if oneaccepts C, then, if one accepts A and B, one must accept Z.”

“Sounds a bit odd, doesn’t it?”“Not really. If A and B together imply Z, then, if both A and B are

true, Z is true. Is there anything odd about that? If so, you can say insteadthat, if A and B are true, Z is true—provided A and B imply Z. Andthere’s nothing odd about that! That’s just ordinary standard English!”

“And I am to take it that this has got something to do with our argu-ment?”

“Of course! You look at it like this: by making C an extra premissand at the same time making the statement of the new implication afirst order hypothetical, you have assumed that the hypothetical C canbe included in its own protasis!”

“I have said, so to speak, that if A and B are true and that if it is truethat if A and B are true Z is true, then Z is true?”

“That’s right. And by making C an indispensable premiss you havefurther assumed that this hypothetical cannot be true unless it is in-cluded in its own protasis!”

14 W. J. REES


“But in that case, . . .” said the Tortoise.“In that case,” said Achilles, “the hypothetical in the protasis must

also be true and included in its own protasis, and so on, ad infinitum.Which means that C can never be true, although you have already ac-cepted it as true!”

“The point, I suppose,” said the Tortoise, “is that no hypotheticalcan be included in its own protasis. I’ll take my stand with you on that.But what must I then say? Must I then say that C cannot be an extrapremiss?”

“Not exactly. The trouble is that this word ‘premiss’ is a bit ambigu-ous. It may mean a proposition which, singly or jointly with others ofthe same order, implies a conclusion, or it may mean a propositionwhich implies that some other proposition or propositions of a differentorder imply a conclusion.”

“For convenience,” said the Tortoise, “let’s call the first a premiss,and the second a meta-premiss. This is going to mean generally, ofcourse, that the major premiss in what generations of logicians aregoing to call a hypothetical argument, will not be a premiss but a meta-premiss—but I suppose that is by the way.”

“Quite so,” said Achilles. “The point now is that C cannot be an extrapremiss, but if by a premiss you mean a meta-premiss, then you canhave C, if you like, as a premiss.”

“Most confusing!” said the Tortoise. “Tell me this: you distinguishbetween the premisses of an inference and the principle of an infer-ence?”

“Most certainly! No reader of Joseph and Russell—once they havewritten on the matter, of course—can object to that distinction.”

“Very well then,” said the Tortoise. “Are you now saying that thepremiss of an inference is a first order statement, while the principle ofan inference is a higher order statement, a statement about certain otherstatements, a meta-statement?”



“I am saying that much, certainly. We must distinguish between anargument, which is an operation with certain statements, and the prin-ciple of an argument, which is a statement about that operation.”

“And, I suppose, the principle of an argument cannot itself be in-cluded among those other statements which are asserted in that argu-ment? It cannot be asserted, in the way in which those other statementsare asserted?”

“That is right. At least, not without destroying the argument, or elsecreating a new kind of argument.”

“Such delightful sophistry!” murmured the Tortoise. “If I take tofootball, you will have to take to professional philosophy!”

“Is that a kind of game?” enquired Achilles.“Of course,” said the Tortoise. “It’s a game in which everybody con-

fuses everybody else, and the winner is the one who confuses most,and all the participants are Players and not Gentlemen.”

“Then I will be a Gentleman,” said Achilles.“In that case,” said the Tortoise, “you must give a straight answer.

Are you, or are you not, saying that the principle of an inference cannotbe an additional premiss of that inference?”

“I am saying that,” said Achilles, “and more. The principle of an in-ference cannot be an extra premiss of the same inference. But that doesn’tmean that it cannot be an extra premiss. What it does mean is that ifyou make it an extra premiss, then the inference becomes a differentkind of inference, your new premiss will be a different kind of premiss,namely a meta-premiss, and the principle of this new inference will bea different kind of principle, namely one which can be stated only bymeans of a second order hypothetical.”

“Good!” said the Tortoise. “I will now have C as a meta-premiss,and I will have the new implication statement as a second order hypothetical.”

16 W. J. REES


“Excellent!” said Achilles. “You are now on the right road. Proceed!”“Suppose,” continued the Tortoise, “I reject this second order hypo-

thetical. Will you not then have to put it also among the premisses—sorry, meta-premisses—and so generate third, fourth, fifth orderhypotheticals, and so on, ad infinitum? If so, you still haven’t presentedme with sufficient premisses, that is, meta-premisses, to enable me todeduce the truth of Z.”

“Now, my dear Tortoise,” said Achilles, “I must ask you to noticejust one further point. A meta-premiss differs from a premiss in oneimportant respect, that only one meta-premiss is ever necessary in anyargument. For if one such meta-premiss is true, the premisses referredto in that meta-premiss are in fact sufficient to imply a conclusion. Anyadditional meta-premisses therefore are quite superfluous.”

“Either C is not a premiss,” said the Tortoise, recapitulating, “or ifit is a premiss it is a meta-premiss, in which case one such premiss issufficient in any argument. . . .”

“That is so,” said Achilles, putting his note-book and his pencil backin his helmet—for pockets had not been invented in those days. “AndI dare say,” he continued, “that you will now find that you have suffi-cient premisses to enable you to deduce the truth of Z.”

At this point the narrator, having some pressing business at the Bank,was obliged to leave. He overheard Achilles say, however, somethingor other about the off-side rule and the advisability of ‘not handling theball unless you are the goalkeeper’. The Tortoise replied that philoso-phy was ‘nowadays a much better game than football’. On that mostcharming note the narrator drove off, and left the happy pair.



3

What Achilles Should Have

Said to the Tortoise

J. F. Thomson

In 1895 Lewis Carroll published in Mind1 a brief dialogue, ‘What theTortoise said to Achilles’. The intention of the story is, plainly enough,to raise a difficulty about the idea of valid arguments, a difficulty sim-ilar, or so Carroll implies, to Zeno’s difficulty about getting to the endof a race-course. Different writers have said different things, usuallybriefly, about what the difficulty is. Let us first consider just what hap-pens in the story and then try to see what problems it raises.

The topic of the story is a certain task set to Achilles. The Tortoisesays that there might be someone who accepted the two propositions:

(A) Things that are equal to the same are equal to each other, and(B) The two sides of this triangle are things that are equal to the

same,

but did not accept

(C) If things that are equal to the same are equal to each other, and ifthe two sides of this triangle are equal to the same, then the twosides of this triangle are equal to each other.

19


Such a person, he says, would not as yet (Lewis Carroll’s italics) beunder any logical necessity to accept the consequent of C, namely

(Z) The two sides of this triangle are equal to each other.

He asks Achilles to pretend that he is such a person and to force him,‘logically’, to accept Z. Since it is Achilles’ failure to do this which isthe point of the story, we must ask how his failure comes about.

Achilles sets about his task in an unexpected way. You might expecthim to begin by trying to find out why the Tortoise does not accept C.Instead, he asks him to accept C, i.e. asks him to accept that very propo-sition which he has just said he does not accept. You might now expectthe Tortoise to laugh, or to be surprised, or at least to say: ‘But I don’taccept C, or so we are pretending.’ Instead he grants the request, orsays he does, and for no other reason than that he has been asked to.He is now on record as having accepted A, B, and C. And now, Achillessays, he must accept Z. ‘If you accept A and B and C, you must acceptZ . . . because it follows logically from them.’ The Tortoise replies ineffect that just as there might be someone who did not accept the hy-pothetical C which connects A and B with Z so there might be someonewho did not accept the hypothetical, call it D, which connects A and Band C with Z, and that such a person might accept each of A and B andC and still not accept Z. When Achilles asks him to accept D he doesso, just as he accepted C, and the story goes on as before. Apparentlythe end of this ‘ideal race-course’ is never to be reached.

To see clearly what is happening, let us relabel the propositions in-volved. The original premises A and B we can collapse into one con-junctive premise C0. The first hypothetical C we shall sometimes call‘C1’, the second, D, ‘C2’, and so on. The hypothetical with antecedentX and consequent Y we shall call ‘X→Y’. Then the sequence of propo-sitions successively accepted by the Tortoise is

20 J. F. THOMSON


C0 = (A & B)C1 = C0 → Z

C2 = (C0 & C1) → ZC3 = (C0 & C1 & C2) → Z

etc. The sequence is generated by the rule that the first term is (A & B)and each term thereafter is the hypothetical whose antecedent is theconjunction of the preceding terms and whose consequent is Z. The be-haviour of the Tortoise also follows a simple inductive rule. He acceptsthe first term of the sequence. At each stage thereafter, having acceptedC0, C1, . . ., Cn, he refuses to accept Z on the grounds that he has notyet accepted Cn+1, is asked to accept this one, does so, and the gamegoes on as before.

It is plain that as long as this procedure is adopted he will never bebrought to accept Z. If at every stage Z is not to be accepted until someother proposition is accepted, Z will never be accepted. But the sensiblereader will ask: ‘So what?’ Why should this procedure be adopted inthe first place? How does, why should, an infinite sequence of hypo-theticals C1, C2, . . . get into the picture?

The Tortoise represents himself as someone who accepts A and Bbut not C and he says that, being in this position, he is not as yet underany logical necessity to accept Z. This is wrong. Whether or not he ac-cepts C, it is logically true. That means that the argument from A andB to Z is logically valid and that the Tortoise in accepting A and B com-mits himself to accepting Z. So he is already under a logical necessityto accept it. To say that he is not (‘as yet’) is precisely to deny that theargument ‘A, B, ∴Z’ is logically valid. But if that were true there wouldbe no problem; we should not expect Achilles to be able logically toforce the Tortoise to accept Z on the basis of an invalid argument. Itmay be objected that the Tortoise is justified ‘from his own point ofview’ in saying that he can accept A and B without accepting Z. The

3 What Achilles Should Have Said to the Tortoise 21


reply is that this point of view is a mistaken one and Achilles’ task isprecisely to make him give it up. How can he do that? He must firstfind out why the Tortoise does not accept C. Someone who was reallyunwilling or unable to grant the truth of this proposition would eitherhave some reason, perverse or ingenious or both, for thinking it falseor doubtful, or he would not have considered sufficiently carefully justwhat proposition it is. Perhaps there are other possibilities. But anywayAchilles must ask the Tortoise to show at least part of his hand. If thelatter’s pretence not to see that C is true is to be considered at all it mustbe taken seriously.

What Achilles does in the story is quite different. In effect he says:‘So you don’t accept C. Well then, will you accept C?’ To make such arequest in such circumstances is ridiculous, and to accede to it is ridicu-lous too. Achilles makes it because, as he himself says, if you accept Aand B and C you must accept Z—‘it follows logically from them’. Butthis is a bad reason. In saying that Z follows from A and B and CAchilles implies that it does not follow from A and B alone, he impliesthat these premises are not by themselves sufficient. He thus acceptsthe implications of the ‘not as yet’ and so makes a nonsense of his ac-ceptance of the idea that he should (‘logically’) force the Tortoise toaccept Z. And anyway in so far as the latter is justified, ‘from his ownpoint of view’, in not accepting Z, he would be justified also in refusingto accept C. He could say: ‘Of course if I accept C I shall then have toaccept Z, but that is not in question. You are trying to get me to acceptZ. You can do that by presenting me with an argument which I see tobe valid and which has premises which I am able to accept. I don’t ac-cept that the argument “A, B, ∴Z” is valid. So, as you should haveforeseen, I can’t accept C. So I can’t accept all the premises of yoursecond argument, “A, B, C, ∴Z”. From the point of view of establish-ing its conclusion, a valid argument with false premises is no better off

22 J. F. THOMSON


than an invalid one. In your case the falsity of the false premise in thesecond argument follows directly from the invalidity of the first. So notonly does the second argument give me no more reason to accept Zthan the first one did, but there is just the same lack of reason in eachcase.’

Given an argument with premises P1, P2, . . . Pκ and conclusion Qlet us call (P1, & P2 & . . . & Pκ)→Q the hypothetical associated withthat argument, and let us call the argument with the same conclusionand premises P1, P2, . . ., (P1 & . . . & Pκ)→Q the strengthened form ofthe original argument and a strengthened argument. An argument mayfail to establish its conclusion on either or both of two counts; it mayhave one or more false premises, and, independently, the relation re-quired to hold between the premises and the conclusion may not hold.It is clear that a strengthened argument will always be valid and so willnever fail on the second count, and that if an argument fails on anycount its strengthening must fail on the first of them. In particular, ifan argument fails by not having enough premises its strengthening willescape that weakness but must, just because it is the strengthened formof that argument, fail by having an unacceptable premise. It followsthat from the point of view of getting arguments which establish theirconclusions the operation of strengthening is either redundant or futile.

We need not be inhibited from accepting this by feelings of loyaltyto the old idea that some arguments have suppressed premises. Cer-tainly, if the argument ‘P, ∴Q’ can have a suppressed premise, whyshould it not have the suppressed premise P→Q? About the idea thatarguments do have suppressed premises, a good deal needs to be said,but it does not need to be said here. For arguments which are said tohave suppressed premises are said to be valid in virtue of having them,and valid arguments do not need to be strengthened. In other words, ifwe wish to say that an argument has suppressed premises we must take



this seriously and really count the suppressed premises among its prem-ises. (The observation that strengthening is either redundant or futileis quite independent of the question whether ‘valid’ must always mean‘logically valid’ or whether logically valid arguments are just a sub-class of valid arguments.)

All this, then, or part of it, is what the Tortoise could have said inreply to Achilles’ request that he accept C. Instead, he accedes to therequest and still does not accept Z. But is this inability the old inabilityin a new guise or is it a new one? Whether Lewis Carroll realized thefact or not, it is a new one. The failure to see the truth of C is, roughlyspeaking, a failure to appreciate the transitivity of the relation samenessof length. The failure to see that C3 is true is a failure to appreciate thelogical force of if. If there could be someone who thought that C1 wasfalse or doubtful, he might well be, and probably would be, someonewho at once saw the truth of C2. So at this second stage of the gamethe Tortoise has changed his ground. He began by representing himselfas someone who could not accept a certain hypothetical. He now pre-tends to have accepted that hypothetical and represents himself assomeone who does not see the truth of a quite different hypothetical.(And in a moment he will change his ground yet again; he will pretendto accept C2 and will make difficulties over C3. But these later sub-terfuges are not very interesting.)

We now see how the infinite sequence of hypothetical gets into thestory. When he says that someone who accepts A and B but not C isnot as yet under any necessity to accept Z, the Tortoise implies not onlythat the premises A and B are not sufficient but also that A and B andC would be, and also that he sees that this is so. When having acceptedC he shifts his difficulty to C2 he implies that if only he were able toaccept that one he would be able to accept Z. So at each stage he intro-duces a new hypothetical into the discussion and tempts Achilles to ask

24 J. F. THOMSON


him to accept it. The sequence of hypotheticals introduced in this wayis infinite because however many premises he accepts he pretends notto see that there are enough.

We have also answered the question why Achilles fails in his task.His first mistake is in asking the Tortoise to accept C. By doing this heimplies that he is not after all in any position to force him, logically, toaccept Z. But if we think that his failure is a punishment for that mis-take, we must be clear that the punishment does not fit the crime. Forthe Tortoise ought not to have acceded to the request and having ac-ceded to it he ought to have accepted Z. So the second thing to be clearabout is that the Tortoise cheats. Instead of presenting Achilles withjust one problem he presents him with infinitely many; though this isconcealed by the fact that Achilles does not really try to solve any ofthem.

The extreme eccentricity of the behaviour of both of the charactersmay well make us wonder whether Lewis Carroll knew what he wasup to in writing the story. Certainly it cannot be merely taken forgranted that he intended to advance some moderately clear thesis ortheses about inference but chose to do so in a veiled and cryptic way.It is just as likely that the story is the expression of a perplexity bysomeone who was not able to make clear to himself just why he wasperplexed. But we may still ask what points of logical interest emergefrom it. I shall mention just two.

We say that if a triangle is isosceles the angles at the base must beequal, that if Tom is older than Dick and Dick older than Harry thenTom must be older than Harry. More generally we say that if such-and-such it must be the case that so-and-so. This use of ‘must’ is a signalthat something is being claimed to follow from something else. But wealso say: if you accept that such-and-such then you must accept thatso-and-so. This use of ‘must’ can be misunderstood. ‘I accept A and B



and C and D’, says the Tortoise at one point. ‘Suppose I still refuse toaccept Z?’ ‘Then Logic would take you by the throat and force you todo it’, Achilles replies. But Logic does no such thing.

‘If you accept the premises of a logically valid argument, you mustaccept its conclusion.’ Well, why must he?—This statement does notmean that if someone does accept the premises of such an argument hewill accept its conclusion, let alone that he will necessarily accept it.He may accept the premises without knowing or without noticing thatthey are the premises of a logically valid argument with that conclusion.Even when the argument is put before him he may be unable to under-stand it or unwilling to try. Or he may not see that it is valid, or maythink that it contains such-and-such a fallacy. He may even say: ‘Sincethe premises are true and the conclusion false the argument must befallacious, though I can’t for the moment see where the fallacy is.’ Evenwhen he has seen and examined the argument and convinced himselfthat it is valid he may still not accept the conclusion, since he may pre-fer to retract his acceptance of the premises. What is true is that in ac-cepting the premises he commits himself to acceptance of theconclusion. Why? Because what we are here calling the conclusion issomething that follows from premises which he accepts. But why thendoes acceptance of a set of premises commit one to acceptance of theirconsequences? This question can be regarded only as a request for anexplanation of the notion of a consequence and of a logically valid ar-gument or as an occasion to remind someone of what these notions are.Part of this explanation is that the set consisting of the premises of alogically valid argument and the negation of its conclusion is logicallyself-inconsistent and so must contain at least one falsehood. So anyonewho accepts the premises and denies the conclusion has committedhimself to at least one falsehood. This is the threat behind the ‘must’.‘If you assert the premises and deny the conclusion, you will have saidat least one false thing, however the facts may turn out to be.’

26 J. F. THOMSON


‘If you accept these propositions you must accept that one’—this ischaracteristically said by someone who is trying to get his hearer to ac-cept something. So it is said by someone who is or has been arguing.Then we may suppose that an argument has been put forward and thatthe hearer is or has been or soon will be examining it. But when thespeaker says what he says he is only saying that the argument is valid.It follows that although this remark is typically made by someone whois arguing it is not itself a piece of an argument. It is one thing to putforward an argument, even a valid one, and another to say that you arearguing validly. It is one thing to propose for acceptance propositionswhich (you hope or believe or know) entail another proposition andanother thing to say that they do. In arguing, you may need to point outthat you are. You then (as it were) step aside from what you are doingand comment on your own performance. But then the performancemust be there, independently of the comment, to be commented on.

The proposition that such-and-such an argument is valid can itselfbe a premise of an argument2. But it cannot be a premise in the argu-ment to which it refers. If you want to say of some argument that it isvalid you must be able to say what argument it is that you want to makethis claim for. The argument must be identifiable. And the identificationmust be such as to allow the claim that it is logically valid to be as-sessed. To assess that claim we need to know what the premises areand what the conclusion is. So the premises must be identifiable inde-pendently of the claim that there are enough of them.

What has just been said about the statement that the argument‘P1, . . ., Pn, ∴Q’ is logically valid must hold also of the statement thatif P1 and P2 and . . . and Pn then necessarily Q. For the latter statementis logically equivalent to the former. It does not matter that the formerargument is explicitly about an argument and the latter not. Just as thestatement that an argument is logically valid cannot turn out to be apremise in that argument, so, and indeed very obviously, a hypothetical



cannot turn out to be its own antecedent or a conjunct in its own an-tecedent. So if, having got you to accept premises P1 to Pn and wantingyou now to do what I think you are committed to doing, viz. accept Q,I assert that if P1 and . . . and Pn then necessarily Q, I am not, or shouldnot regard myself as, asking you to accept another premise. For ex hy-pothesi I suppose that you already have enough premises.

To say this is not to deny that some arguments have hypotheticalsas premises and have them as premises in just the way they have otherpremises3. Someone who, having put forward some premises, puts for-ward a hypothetical having the conjunction of those premises as its an-tecedent may very well intend the hypothetical to be counted as anotherpremise. If what is in question is the validity of your argument, it is upto you to say what its premises are. You may list as the set of premisesenough to make it logically valid, and you may, either knowingly orunwittingly, list some that are redundant. All that is being said is that ifyou list your premises and all of your premises and then assert whatwe called the hypothetical associated with the argument whose prem-ises these are, that hypothetical just cannot turn out to be one of thepremises already listed. This rests on the fact that a hypothetical cannotbe a conjunct in its own antecedent, and this rests in turn on the factthat no sentence which expresses a proposition can be longer than it is.It is therefore very obvious. But it is enough to clear up one of the mis-understandings in the story. When Achilles said that if you accept Aand B and C you must accept Z he was claiming that the argument thatsince A, B, and C, therefore Z was logically valid, had enough prem-ises, and so was not, or should not have regarded himself as, offeringanother premise.

So the first point of interest is that we must distinguish between ar-guing and talking about an argument, between giving reasons, evengood ones, and saying that some reasons are good ones. In particular,

28 J. F. THOMSON


if someone in arguing asserts a hypothetical and makes it clear, by usingsome such signal-word as ‘must’ or ‘logically’ or ‘necessarily’, that heregards it as necessarily true, he may be offering a premise and he maybe doing something equivalent to commenting on a set of premises al-ready identifiable. What he cannot be doing is both at once.

The second point is connected with the first. Before we can hope tounderstand what is going on between Achilles and the Tortoise we mustbe clear that to assert the truth (logical truth, or acceptability, or rea-sonableness, etc.) of a hypothetical is equivalent to asserting the valid-ity (logical validity, or cogency, etc.) of the argument with which thathypothetical is associated. It follows that to accept the hypothetical isto commit oneself to accepting the validity of the argument. But whatis it to accept the validity of an argument? One thing that shows thatyou accept it is that if you assert the premises you are willing to go onand say ‘therefore’ and then assert the conclusion. But then supposethat someone claims to accept the hypothetical and to accept the prem-ise but is not willing to assert the conclusion? How can we get himactually to do what he is committed to doing, i.e. accept Q? It is naturalto think of pointing out to him that Q follows logically from P andP→Q, and this thought may then seem suspect for something like thefollowing reason: we began by wanting him to accept the argument‘P, ∴Q’ and now seem to be trying to get him to accept the (different)argument ‘P, P→Q ∴Q’; what if he will not accept this one either, shallwe then have to start again? The suspicion is dispelled when we reflectthat the latter argument really is different from the former one, so thatsomeone might accept it and not accept the former. We must also re-member that when we claim validity for the latter argument we are not,or at least should not regard ourselves as, arguing that since it is validso is the original one. Such an argument would be fallacious. Strength-ened arguments are always valid. So the second point of interest is that



logically valid arguments are of different kinds. Consider for examplethe three arguments ‘A, B, ∴Z’, ‘A, B, C, ∴Z’ and ‘B, ∴Z’. The firstis formalizable in first-order predicate logic with identity. The secondis formalizable in truth-functional logic and in any one of a large num-ber of weaker systems of propositional logic. The third, though logi-cally valid, is not formally valid at all.

We naturally feel a reluctance to admit that someone could acceptA, B and C and not accept Z. Behind this is the fact that if someoneclaims to accept the premises of a very simple argument and does notaccept the conclusion it is sometimes reasonable to suppose that he hasnot really accepted the premises4. That is, we sometimes make it a nec-essary condition for someone’s having accepted a set of propositionsthat he accepts such-and-such consequences of them. No general rulescan be given for when this is reasonable, but it is probably a mere prej-udice to think that the difficulty arises especially over ‘A, B, C, ∴Z’and does not arise at all over ‘A, B, ∴Z’. But the important point isthat it is not the Tortoise’s refusal to accept Z at the second stage thatshows that Achilles was silly to offer him C as a premise at the firststage; even though Achilles was silly to do so, for reasons we haveseen. What that refusal shows is rather something about the Tortoise.

In conclusion I should like to comment briefly on some remarksabout the story in Professor G. Ryle’s paper If, So, and Because5. Ryleis here considering the question: How does the validity of the argument‘P, ∴Q’ require the truth of the hypothetical P→Q? He discussesamong others the following answer: ‘The argument is always invalidunless the premise from which Q is drawn incorporates not only P butalso P→Q. Q follows neither from P→Q by itself, nor from P by itself,but only from the conjunction P and (P→Q).’ Ryle comments on thisidea as follows: ‘But this notoriously will not do. For, suppose it did.Then a critic might ask to be satisfied that Q was legitimately drawn

30 J. F. THOMSON


from P and (P→Q); and to be satisfied he would have to be assured thatif P and also if P and Q then Q. So this new hypothetical would haveto be incorporated as a third component of a conjunctive premise, andso on for ever—as the Tortoise proved to Achilles. The principle of aninference cannot be one of its premises or part of its premise. Conclu-sions are drawn from premises in accordance with principles, not frompremises which embody those principles.’

It seems that what Ryle calls the principle of an inference is eitherwhat we have called the hypothetical associated with the argument orsome statement or formula of which that hypothetical is an exemplifi-cation or a general proposition of which the hypothetical is a particularcase. In each of these cases his statement that the principle cannot beone of the premises or part of its premise is clearly correct. It is hardlynecessary to repeat the argument: the ‘principle’ of the argument ‘A,B, ∴Ζ’ is, roughly speaking, the principle that a certain relation is tran-sitive; if we strengthen that argument by adding the appropriate hypo-thetical as a redundant premise the new argument has a quite differentprinciple. But, more or less clearly implicit in what Ryle says, there isthe suggestion that Achilles fails in his task because he does not distin-guish premises from principles, and, coupled with it, the idea that thenecessity for this distinction can be demonstrated by means of aregress-argument. This does not seem correct.

We must notice first that the suggestion which Ryle is attacking ismuch more seriously confused than his comment on it brings out. Forhow in it are the letters ‘P’ and ‘Q’ being used? If they are constantswe can hardly be expected to assess the idea that the argument ‘P, ∴Q’is not as it stands valid, since we have not been told what propositionsP and Q are. But if they are variables, the suggestion comes to this: noargument is valid, but, given an argument, which will of course be in-valid, we can always obtain from it an argument (its strengthened form)



which will be valid. And while it is absurd to hold that no argumentsare valid, it is doubly absurd to hold this and then say that some argu-ments can be made valid. If for no values of ‘P’ and ‘Q’ does P yieldQ, then, in particular, P & (P→Q) does not yield Q, since P & (P→Q)is just one value of ‘P’.

So, to dismiss the suggestion, we need only be clear what it comesto, and we do not need to invoke a regress-argument. But it is not cleareither that we are entitled to do so. The suggestion that we are dependsupon thinking that if someone cavils at the argument ‘A, B, ∴Z’ on thegrounds that C is not one of its premises he is somehow committed tocavilling at ‘A, B, C, ∴Z’ because C2 is not one of its premises. But thisis just wrong. If someone had a prejudice in favour of truth-functionallyvalid arguments he would be consistent in rejecting the first argumentas invalid and then accepting the second. A critic who then asked to besatisfied that Z was legitimately drawn from A, B, and C would beshown a truth-table and that would be that. So there is no force in Ryle’ssuggestion that ‘this new hypothetical (here, C2) would have to be in-corporated as a third component of a conjunctive premise’.

If all this is correct, then what is most usually taken to be establishedby the story, namely that we must not try to make the ‘principle’ of aninference one of its premises, on pain of running into an infinite regress,is wrong, and is not established by the story. What people who say thismean by taking the principle as one of the premises turns out to be whatwe called strengthening, and strengthening does not run us into aregress. The mistake of supposing that it does comes partly from failingto notice that the Tortoise changes his ground, shifts his difficulty, atthe second stage. It is true that if someone thinks that every argumentneeds to be strengthened he will think or be committed to thinking thatevery argument is invalid, but to expose this we do not need to invokea regress argument anyway. Neither does any such argument help us

32 J. F. THOMSON


in seeing what needs to be seen, the way in which strengthening is ei-ther redundant or futile. The infinite regress is just an infinitely longred herring.

NOTES

1. New Series, vol. IV, pp. 278–80.2. For example, an argument designed to show that such-and-such a book

contains exactly one valid argument.3. At least one writer on the story has been led to deny this. See D. G.

Brown, ‘What the Tortoise taught us’, Mind, vol. LXIII (1954), p. 179.4. See the paper cited in the previous footnote.5. In ‘Philosophical Analysis’ edited by Max Black, New York, 1950.



QUESTIONS

I. LOGIC AND KNOWLEDGE

1. The tortoise asks Achilles to “force me, logically, to accept Z as true,”but can anyone be logically forced to accept a proposition?

2. Does Rees’s distinction between premises and meta-premises solve theproblem of infinite regress?

3. Is Thompson correct that accepting a proposition is also agreeing toperform specific inferences from that proposition?

35


II

LOGIC AND

DEFINITION

The logical connectives—such as and, or, and if/then—can bedefined semantically in terms of a truth table or syntactically byhow they function within the rules of inference, such as modusponens and modus tollens. The semantics and syntax for the con-nectives are supposed to complement each other.

But consider a new connective—tonk. We don’t know the se-mantics for tonk, but it is governed by two familiar rules of in-ference: one analogous to the Addition rule (p entails p or q),the other analogous to the Simplification rule (p and q entailsq). A. N. Prior explores the difficulties with any connectivewhose syntax is not constrained by semantics. J. I. Stevensonargues that any set of rules that define a connective must pass asemantic test of “complete justification.” Nuel D. Belnap claimsthat the requirements provided by the semantic test can be cap-tured by additional syntactic rules.


4

The Runabout Inference-Ticket

A. N. Prior

It is sometimes alleged that there are inferences whose validity arisessolely from the meanings of certain expressions occurring in them. Theprecise technicalities employed are not important, but let us say thatsuch inferences, if any such there be, are analytically valid.

One sort of inference which is sometimes said to be in this sense an-alytically valid is the passage from a conjunction to either of its con-juncts, e.g., the inference ‘Grass is green and the sky is blue, thereforegrass is green.’ The validity of this inference is said to arise solely fromthe meaning of the word ‘and.’ For if we are asked what is the meaningof the word ‘and,’ at least in the purely conjunctive sense (as opposedto, e.g., its colloquial use to mean ‘and then’), the answer is said to becompletely given by saying that (i) from any pair of statements P andQ we can infer the statement formed by joining P to Q by ‘and’ (whichstatement we hereafter describe as ‘the statement P-and-Q’), that (ii)from any conjunctive statement P-and-Q we can infer P, and (iii) fromP-and-Q we can always infer Q. Anyone who has learnt to performthese inferences knows the meaning of ‘and,’ for there is simply nothingmore to knowing the meaning of ‘and’ than being able to perform theseinferences.

39


A doubt might be raised as to whether it is really the case that, forany pair of statements P and Q, there is always a statement R such thatgiven P and given Q we can infer R, and given R we can infer P andcan also infer Q. But on the view we are considering such a doubt isquite misplaced, once we have introduced a word, say the word ‘and,’precisely in order to form a statement R with these properties from anypair of statements P and Q. The doubt reflects the old superstitious viewthat an expression must have some independently determined meaningbefore we can discover whether inferences involving it are valid or in-valid. With analytically valid inferences this just isn’t so.

I hope the conception of an analytically valid inference is now atleast as clear to my readers as it is to myself. If not, further illuminationis obtainable from Professor Popper’s paper on ‘Logic without As-sumptions’ in Proceedings of the Aristotelian Society for 1946–7, andfrom Professor Kneale’s contribution to Contemporary British Philos-ophy, Volume III. I have also been much helped in my understandingof the notion by some lectures of Mr. Strawson’s and some notes ofMr. Hare’s.

I want now to draw attention to a point not generally noticed, namelythat in this sense of ‘analytically valid’ any statement whatever may beinferred, in an analytically valid way, from any other. ‘2 and 2 are 5,’for instance, from ‘2 and 2 are 4.’ It is done in two steps, thus:

• 2 and 2 are 4.• Therefore, 2 and 2 are 4 tonk 2 and 2 are 5.• Therefore, 2 and 2 are 5.

There may well be readers who have not previously encountered thisconjunction ‘tonk,’ it being a comparatively recent addition to the lan-guage; but it is the simplest matter in the world to explain what it

40 A. N. PRIOR


means. Its meaning is completely given by the rules that (i) from anystatement P we can infer any statement formed by joining P to any state-ment Q by ‘tonk’ (which compound statement we hereafter describe as‘the statement P-tonk-Q’), and that (ii) from any ‘contonktive’ state-ment P-tonk-Q we can infer the contained statement Q.

A doubt might be raised as to whether it is really the case that, forany pair of statements P and Q, there is always a statement R such thatgiven P we can infer R, and given R we can infer Q. But this doubt isof course quite misplaced, now that we have introduced the word ‘tonk’precisely in order to form a statement R with these properties from anypair of statements P and Q.

As a matter of simple history, there have been logicians of some em-inence who have seriously doubted whether sentences of the form ‘Pand Q’ express single propositions (and so, e.g., have negations). Aris-totle himself, in De Soph. Elench. 176 a 1 ff., denies that ‘Are Calliasand Themistocles musical?’ is a single question; and J. S. Mill says of‘Caesar is dead and Brutus is alive’ that ‘we might as well call a streeta complex house, as these two propositions a complex proposition’(System of Logic I, iv. 3). So it is not to be wondered at if the form ‘Ptonk Q’ is greeted at first with similar scepticism. But more enlightenedviews will surely prevail at last, especially when men consider the ex-treme convenience of the new form, which promises to banish falscheSpitfizndigkeit from Logic for ever.

4 The Runabout Inference-Ticket 41


5

Roundabout the Runabout

Inference-Ticket

J. T. Stevenson

In his article “The Runabout Inference-Ticket” Professor A. N. Priortries to show that there is an absurdity derivable from the theory “. . .that there are inferences whose validity arises solely from the meaningsof certain expressions occurring in them.”1 For accounts of the theoryother than his own Prior refers us to the writings of K. R. Popper, W.Kneale, P. F. Strawson, and R. M. Hare. I shall not be concerned to de-termine whether he has accurately represented their versions of the the-ory (although I think this doubtful for example in the case of Popper),because Prior’s interpretation is itself intrinsically interesting. Prior’sargument strongly suggests that there is something wrong with the the-ory, as he presents it, but, unfortunately, he does not show us what iswrong with it. I wish to show (1) exactly where the theory, as stated byPrior, goes wrong, and (2) that the theory can be stated in such a wayas to be quite sound.

According to the theory in question, the inference ‘Grass is greenand the sky is blue, therefore grass is green’ is an analytically valid in-ference solely in virtue of the meaning of the word ‘and.’ The presumedanalyticity of the inference is exhibited by the following statement ofthe meaning of the word ‘and’: “. . . (i) from any pair of statements P

43


and Q we can infer the statement formed by joining P to Q by ‘and’ . . .(ii) from any conjunctive statement P-and-Q we can infer P, and (iii)from P-and-Q we can always infer Q.”2

Prior attempts to reduce the foregoing theory to absurdity by intro-ducing a new connective ‘tonk’ and giving it a meaning in the way sug-gested by the theory. The complete meaning of ‘tonk’ is: “(i) from anystatement P we can infer any statement formed by joining P to any state-ment Q by ‘tonk’ . . ., and . . . (ii) from any ‘contonktive’ statement P-tonk-Q we can infer the contained statement statement Q.”3 He thenshows that the following inference is valid in virtue of the meaning of‘tonk’:

2 and 2 are 4.Therefore, 2 and 2 are 4 tonk 2 and 2 are 5.Therefore, 2 and 2 are 5.4

Prior does not say, but seems to imply, this: Since the theory allows todeduce, “in an analytically valid way”, a patently false statement froma patently true one, there must be something radically wrong with thetheory.

In order to understand what has happened here, it is essential to no-tice that the theory requires us to give the meaning of logical connec-tives in terms of rules. These rules are permissive: I take it that theforce of ‘we can infer,’ as it occurs in the foregoing definitions, is thesame as ‘we may infer’ or ‘we are allowed or permitted to infer.’ If ‘wecan infer’ were taken to mean the same as ‘we may validly infer,’ someof the things I shall say would have to be modified. But, in this case, if‘valid’ were used in its ordinary sense (namely, such as to lead fromtruth only to truth and never to falsehood), Prior’s definition of ‘tonk’would become radically incoherent, indeed self-contradictory, and hisargument trivially unsound and hence uninteresting. I shall take the

44 J. T. STEVENSON


more interesting and more usual interpretation that rules of inferenceare simply permissive. Granted this, we can now consider two impor-tant insights and one serious error in the theory.

The first insight concerns the meaning of logical connectives: theway in which we can express the meaning of connectives must be dif-ferent from the way in which we express the meanings of non-logicalwords. In the first place, leaving aside Platonism, connectives are notused to denote, and hence the sort of semantical properties they havewill be different from those of non-logical words. Second, logical termsare syncategorematic or incomplete symbols; they have no meaning inisolation. Since the most distinctive feature of a logical term is its syn-tactical properties, we can explain its meaning to someone unfamiliarwith it by exhibiting how these syntactical properties affect the contextsin which the connective in question occurs. And a very convenient wayto do this is to give the permissive rules governing the inferences wecan make using the connective.

The second insight is that we ordinarily justify (i.e., validate) infer-ences by appealing to a permissive rule. If you question my inference‘If I don’t leave in five minutes, I shall be late, and I am not going toleave in five minutes; so I shall be late,’ I justify it by appealing to thepermissive rule modus ponens.

The serious error in the theory consists in combining these two in-sights in an unfortunate way. It is assumed that we can completely jus-tify an inference by appealing to the meaning of a logical connectiveas stated in permissive rules. If this were so, we could, as Prior shows,justify any inference whatsoever by defining a logical connective interms of permissive rules in such a way that we would be allowed topass from true premises to a false conclusion.

The crucial point to be noted is this: in order to completely justifyan inference we must appeal to a sound rule of inference. A completejustification of an inference has two parts: we must first validate the

5 Roundabout the Runabout Inference-Ticket 45


inference by subsuming it under a rule, and secondly we must vindicatethe rule itself by showing that it is a sound rule.5 A deductive rule issound if and only if it permits only valid inferences, an inference beingvalid in this sense if and only if it is such that when the premises aretrue the conclusion must be true. The difficulty in our theory, then, isthat it does not prevent us from defining connectives in terms of un-sound permissive rules. Since no attempt is made to vindicate the rulesused in the definitions, the definitions do not, by themselves, provide acomplete justification of our inferences.

I now turn to the problem of stating the theory in such a way that itavoids the above difficulty. Basically, the theory states that certain in-ferences are completely justified solely in virtue of the meanings as-signed to certain logical connectives. Since giving the meaning of alogical connective consists in giving its syntactical properties, we mustshow that, given a statement of the syntactical properties of a connec-tive, the soundness of certain rules of inference can be demonstrated.There is no difficulty in doing this; it can be done, indeed, it has beendone, for many different connectives, and there is no need to go intodetails here.6

To be more precise, two qualifications should be made. First: thesyntactical properties of a connective include both its formation andtransformation properties, although here only its transformation prop-erties are considered. Second: we can exhibit the transformation prop-erties of a sentence connective in a calculus by making a value-tablefor it either so that the calculus remains uninterpreted, or so that it be-comes interpreted. In the former case, we might use some arbitrarysymbols for the values (say 0 and 1), and deal with pure syntacticalproperties. In the latter case, we use truth and falsity as values; and,since truth is a semantical notion, the calculus becomes to some extentinterpreted, and we are no longer dealing with pure syntactical proper-

46 J. T. STEVENSON


ties. For answering questions of soundness the latter method is the onewhich must be used; but for convenience I continue to speak simply ofsyntactical properties.

In a formal calculus we can state the syntactical properties of, say, atruth-functional binary sentence connective ‘o,’ by stating, in the meta-language, the way in which the truth-value of the well-formed formula‘poq’ is a function of (all possible combinations of) the truth-values ofthe components ‘p’ and ‘q.’ We can then deduce from these statements,in a very rigorous way, a meta-theorem of the calculus (again stated inthe meta-language) to the effect that such-and-such permissive rulesare sound, i.e., lead from truths only to truths and never to falsehoods.Since from a statement of the meaning of a connective we can derivedemonstrably sound permissive rules of inference governing that con-nective, we may say that certain inferences are completely justifiedsolely in virtue of the meanings of certain expressions occurring inthem.

The important difference between the theory of analytic validityas it should be stated and as Prior stated it lies in the fact that he givesthe meanings of connectives in terms of permissive rules, whereasthey should be stated in terms of truth-function statements in a meta-language. The theory of analytic validity does not require that themeanings of connectives be given in terms of rules; as we have seen,to do so is to leave open the question of complete justification. Whatthe correct theory of analytic validity does require is that the meaningsof connectives be given in terms of statements of syntactical proper-ties. When this is done the soundness of certain rules of inference isdemonstrable, and thus inferences can be completely justified by ap-pealing to the meanings of connectives. Using the latter method weblock the introduction of a connective like Prior’s ‘tonk.’ This can beseen as follows.



Consider these two truth-tables which exhibit in a graphic way thesyntactical properties of two binary sentence connectives ‘o’ and ‘§.’

A. p q poq B. p q p§q T T T T T T T F T T F F F T F F T T F F F F F F

RA: p ∴poq RB: p§q ∴q

From A it can be seen, intuitively, that the syntactical properties of ‘o’permit us to demonstrate that the permissive rule RA, namely, from ‘p’you may infer ‘poq,’ is sound; and with a properly formulated statementof these syntactical properties it can be rigorously demonstrated to besound. Similarly, from B it can be seen that the rule RB, namely, from‘p§q’ you may infer ‘q,’ is a sound rule. Prior’s connective ‘tonk’ isgoverned by two rules like RA and RB. The syntactical properties of‘tonk,’ then, must be a combination of the syntactical properties of ‘o’and ‘§’; and in order to demonstrate the joint soundness of the rules for‘tonk,’ we would have to construct a truth-table combining all the fea-tures of A and B. But it is obvious that this would involve ascribingcontradictory syntactical properties to ‘tonk.’ This being so, it wouldbe impossible to state consistently the meaning of ‘tonk’ in the mannerI have suggested; and hence impossible to completely justify the infer-ence from ‘2 and 2 are 4’ to ‘2 and 2 are 5.’ One could, of course, asPrior has done, state the meaning of ‘tonk’ in terms of rules and in thisway justify (i.e., validate) ‘2 and 2 are 4, therefore, 2 and 2 are 5,’ butthis would not completely justify the inference, for it would leave openthe question as to the vindication of the inference. And, of course, bydefinition it could never be vindicated, for it leads from an obvious

48 J. T. STEVENSON


truth to an obvious falsehood. I conclude, then, that there is nothingwrong with the theory of analytic validity when properly stated.7

NOTES

1. Analysis 21.2, December 1960.2. Ibid.3. Ibid.4. Ibid.5. The distinction between validation and vindication is due to H. Feigl.

See “De Principiis non Disputandum . . . ?” in Philosophical Analysis, ed.Max Black (Cornell University Press, 1950).

6. See any standard text, e.g., Church’s Introduction to MathematicalLogic.

7. I should like to acknowledge the benefit I have had of a number ofstimulating discussions with Wesley C. Salmon on this topic. He is not, ofcourse, responsible for any errors I may have made.



6

Tonk, Plonk and Plink1

Nuel D. Belnap

A. N. Prior has recently discussed2 the connective tonk, where tonk isdefined by specifying the role it plays in inference. Prior characterizesthe role of tonk in inference by describing how it behaves as conclusion,and as premiss: (1) A ├ A-tonk-B, and (2) A-tonk-B ├ B (where wehave used the sign ‘├’ for deducibility). We are then led by the transi-tivity of deducibility to the validity of A ├ B, “which promises to banishfalsche Spitzfindigkeit from Logic for ever.”

A possible moral to be drawn is that connectives cannot be definedin terms of deducibility at all; that, for instance, it is illegitimate to de-fine and as that connective such that (1) A-and-B ├ A, (2) A-and-B├B, and (3) A, B ├ A-and-B. We must first, so the moral goes, have anotion of what and means, independently of the role it plays as premissand as conclusion. Truth-tables are one way of specifying this an-tecedent meaning; this seems to be the moral drawn by J. T. Stevenson.3

There are good reasons, however, for defending the legitimacy of defin-ing connections in terms of the roles they play in deductions.

It seems plain that throughout the whole texture of philosophy onecan distinguish two modes of explanation: the analytic mode, which

51


tends to explain wholes in terms of parts, and the synthetic mode, whichexplains parts in terms of the wholes or contexts in which they occur.4

In logic, the analytic mode would be represented by Aristotle, whocommences with terms as the ultimate atoms, explains propositions orjudgments by means of these, syllogisms by means of the propositionswhich go to make them up, and finally ends with the notion of a scienceas a tissue of syllogisms. The analytic mode is also represented by thecontemporary logician who first explains the meaning of complex sen-tences, by means of truth-tables, as a function of their parts, and thenproceeds to give an account of correct inference in terms of the sen-tences occurring therein. The locus classicus of the application of thesynthetic mode is, I suppose, Plato’s treatment of justice in the Repub-lic, where he defines the just man by reference to the larger context ofthe community. Among formal logicians, use of the synthetic mode inlogic is illustrated by Kneale and Popper (cited by Prior), as well as byJaskowski, Gentzen, Fitch, and Curry, all of these treating the meaningof connectives as arising from the role they play in the context of formalinference. It is equally well illustrated, I think, by aspects of Wittgen-stein and those who learned from him to treat the meanings of wordsas arising from the role they play in the context of discourse. It seemsto me nearly self-evident that employment of both modes of explana-tion is important and useful. It would therefore be truly a shame to seethe synthetic mode in logic pass away as a result of a severe attack oftonktitis.

Suppose, then, that we wish to hold that it is after all possible to de-fine connectives contextually, in terms of deducibility. How are we toprevent tonktitis? How are we to make good the claim that there is noconnective such as tonk5 though there is a connective such as and(where tonk and and are defined as above)?

52 NUEL D. BELNAP


It seems to me that the key to a solution6 lies in observing that evenon the synthetic view, we are not defining our connectives ab initio,but rather in terms of an antecedently given context of deducibility, con-cerning which we have some definite notions. By that I mean that be-fore arriving at the problem of characterizing connectives, we havealready made some assumptions about the nature of deducibility. Thatthis is so can be seen immediately by observing Prior’s use of the tran-sitivity of deducibility in order to secure his ingenious result. But if wenote that we already have some assumptions about the context of de-ducibility within which we are operating, it becomes apparent that bya too careless use of definitions, it is possible to create a situation inwhich we are forced to say things inconsistent with those assumptions.

The situation is thus exactly analogous to that, pointed out by Peano,which occurs when one attempts to define an operation, ‘?’, on rationalnumbers as follows:

This definition is inadmissible precisely because it has consequenceswhich contradict prior assumptions; for, as can easily be shown, ad-mitting this definition would lead to (say) .

In short, we can distinguish between the admissibility of the defini-tion of and and the inadmissibility of tonk on the grounds of consis-tency—i.e., consistency with antecedent assumptions. We can give aprecise account of the requirement of consistency from the syntheticpoint of view as follows.

(1) We consider some characterization of deducibility, which may betreated as a formal system, i.e., as a set of axioms and rules involving thesign of deducibility, ‘├’, where ‘A1, . . . , An ├ B’ is read ‘B is deducible

a ? c a + cb d b + d.( ) =df

6 Tonk, Plonk and Plink 53


from A1, . . . , An’ For definiteness, we shall choose as our characteri-zation the structural rules of Gentzen:

Axiom. A├ ARules. Weakening: from A1, . . . , An ├ C

to infer A1, . . . , An B ├ CPermutation: from A1, . . . , Ai, Ai+1, . . . , An ├ B

to infer A1, . . . , Ai+1, Ai, . . . , An ├ B.Contraction: from A1, . . . , An, An ├ B

to infer A1, . . . , An ├ BTransitivity: from A1, . . . , Am├ B

and C1, . . . , Cn, B├ Dto infer A1, . . . , Am, C1, . . . , Cn├ D.

In accordance with the opinions of experts (or even perhaps on moresubstantial grounds) we may take this little system as expressing alland only the universally valid statements and rules expressible in thegiven notation: it completely determines the context.

(2) We may consider the proposed definition of some connective,say plonk, as an extension of the formal system characterizing de-ducibility, and an extension in two senses, (a) The notion of sentenceis extended by introducing A-plonk-B as a sentence, whenever A andB are sentences. (b) We add some axioms or rules governing A-plonk-B as occurring as one of the premisses or as conclusion of a deducibil-ity-statement. These axioms or rules constitute our definition of plonkin terms of the role it plays in inference.

(3) We may now state the demand for the consistency of the definitionof the new connective, plonk, as follows: the extension must be conser-vative7; i.e., although the extension may well have new deducibility-statements, these new statements will all involve plonk. The extension

54 NUEL D. BELNAP


will not have any new deducibility-statements which do not involveplonk itself. It will not lead to any deducibility-statement A1, . . . , An ├B not containing plonk, unless that statement is already provable in theabsence of plonk-axioms plonk-rules. The justification for unpackingthe demand for consistency in terms of conservativeness is precisely ourantecedent assumption that we already had all the universally valid de-ducibility-statements not involving any special connectives.

So the trouble with the definition of tonk given by Prior is that it isinconsistent. It gives us an extension of our original characterizationof deducibility which is not conservative, since in the extension (butnot in the original) we have, for arbitrary A and B, A ├ B. Adding atonkish role to the deducibility-context would be like adding to cricketa player whose role was so specified as to make it impossible to distin-guish winning from losing.

Hence, given that our characterization of deducibility is taken ascomplete, we may with propriety say ‘There is no such connective astonk’; just as we say that there is no operation, ‘?’, on rational numberssuch that On the other hand, it is easily shown thatthe extension got by adding and is conservative, and we may hence say‘There is a connective having these properties.’

It is good to keep in mind that the question of the existence of aconnective having such and such properties is relative to our charac-terization of deducibility. If we had initially allowed A ├ B (!), therewould have been no objection to tonk, since the extension would thenhave been conservative. Also, there would have been no inconsistencyhad we omitted from our characterization of deducibility the rule oftransitivity.

The mathematical analogy leads us to ask if we ought not also toadd uniqueness8 as a requirement for connectives introduced by defi-nitions in terms of deducibility (although clearly this requirement is



not as essential as the first, or at least not in the same way). Suppose,for example, that I propose to define a connective plonk by specifyingthat B ├ A-plonk-B. The extension is easily shown to be conservative,and we may, therefore, say ‘There is a connective having these proper-ties.’ But is there only one? It seems rather odd to say we have definedplonk unless we can show that A-plonk-B is a function of A and B, i.e.,given A and B, there is only one proposition A-plonk-B. But what dowe mean by uniqueness when operating from a synthetic, contextualistpoint of view? Clearly that at most one inferential role is permitted bythe characterization of plonk; i.e., that there cannot be two connectiveswhich share the characterization given to plonk but which otherwisesometimes play different roles. Formally put, uniqueness means that ifexactly the same properties are ascribed to some other connective, sayplink, then A-plink-B will play exactly the same role in inference as A-plonk-B, both as premiss and as conclusion. To say that plonk (charac-terized thus and so) describes a unique way of combining A and B is tosay that if plink is given a characterization formally identical to that ofplonk, then

(1) A1, . . . , B-plonk-C, . . . , An ├ D if and only if A1, . . . , B-plink-C, . . . , An ├ D

and

(2) A1, . . . , An ├ B-plonk-C if and only if A1, . . . , An ├ B-plink-C.

Whether or not we can show this will depend, of course, not onlyon the properties ascribed to the connectives, but also on the propertiesascribed to deducibility. Given the characterization of deducibility

56 NUEL D. BELNAP


above, it is sufficient and necessary that B-plonk-C ├ B-plink-C, andconversely.

Harking back now to the definition of plonk by: B ├ A-plonk-B, itis easy to show that plonk is not unique; that given only: B ├ A-plonk-B, and B ├ A-plink-B, we cannot show that plonk and plink invariablyplay the same role in inference. Hence, the possibility arises that plonkand plink stand for different connectives: the conditions on plonk donot determine a unique connective. On the other hand, if we introducea connective, et, with the same characterization as and, it will turn outthat A-and-B and A-et-B play exactly the same role in inference. Theconditions on and therefore do determine a unique connective.

Though it is difficult to draw a moral from Prior’s delightful notewithout being plonking, I suppose we might put it like this: one candefine connectives in terms of deducibility, but one bears the onus ofproving at least consistency (existence); and if one wishes further totalk about the connective (instead of a connective) satisfying certainconditions, it is necessary to prove uniqueness as well. But it is not nec-essary to have an antecedent idea of the independent meaning of theconnective.

NOTES

1. This research was supported in part by the Office of Naval Research,Group Psychology Branch, Contract No. SAR/Nonr-609(16).

2. ‘The Runabout Inference-ticket,’ Analysis 21.2, December 1960. 3. ‘Roundabout the Runabout Inference-ticket,’ Analysis 21.6, June 1961.

Cf. p. 127: “The important difference between the theory of analytic validity[Prior’s phrase for what is here called the synthetic view] as it should bestated and as Prior stated it lies in the fact that he gives the meaning of con-nectives in terms of permissive rules, whereas they should be stated in termsof truth-function statements in a meta-language.”

4. I learned this way of looking at the matter from R. S. Brumbaugh.



5. That there is no meaningful proposition expressed by A-tonk-B; thatthere is no meaningful sentence A-tonk-B—distinctions suggested by thesealternative modes of expression are irrelevant. Not myself being a victim ofeidophobia, I will continue to use language which treats the connective-word‘tonk’ as standing for the putative propositional connective, tonk. It is equallyirrelevant whether we take the sign ├ as representing a syntactic concept ofdeducibility or a semantic concept of logical consequence.

6. Prior’s note is a gem, reminding one of Lewis Carroll’s ‘What the Tor-toise said to Achilles.’ And as for the latter, so for the former, I suspect thatno solution will ever be universally accepted as the solution.

7. The notion of conservative extensions is due to Emil Post.8. Application to connectives of the notions of existence and uniqueness

was suggested to me by a lecture of H. Hiż.

58 NUEL D. BELNAP


QUESTIONS

II. LOGIC AND DEFINITION

1. According to Prior, what is a runabout inference-ticket?

2. According to Belnap what is tonkitis?

3. Are the laws of logic merely conventional?

59


III

LOGIC AND

INFERENCE

Vann McGee presents what he takes to be a counterexample tomodus ponens. Suppose you are reading The Odyssey. Supposefurther that if you are reading The Odyssey, then if you are read-ing the first book, you are reading about Telemachus and Pene-lope. These premises seem to entail that if you are reading thefirst book, then you are reading about Telemachus. Yet this con-clusion may not be true. You could be reading the first book ofthe Iliad, and so you are reading about Achilles and Brisias; oryou might be reading the first book of the Bible, and so youwould be reading about Adam and Eve.

E. J. Lowe suggests that “if” clauses are intrinsically ambigu-ous in the English language and that McGee’s proposed counter-example can be defused with a more careful translation from theEnglish sentences into logical notation. D. E. Over contends thatthe context of statement, if you are reading the first book, thenyou are reading about Telemachus, makes clear that the statementis true of the specific book you are reading and not just any book.


7

A Counterexample to Modus Ponens1

Vann McGee

The rule of modus ponens, which tells us that from an indicative con-ditional ┌If φ then ψ┐,2 together with its antecedent φ, one can inferψ, is one of the fundamental principles of logic.3 Yet, as the followingexamples show, it is not strictly valid; there are occasions on which onehas good grounds for believing the premises of an application of modusponens but yet one is not justified in accepting the conclusion. Lateron, we shall see how these examples can be modified to give coun-terexamples to Stalnaker’s semantics for the conditional:

Opinion polls taken just before the 1980 election showed the Re-publican Ronald Reagan decisively ahead of the Democrat JimmyCarter, with the other Republican in the race, John Anderson, a dis-tant third. Those apprised of the poll results believed, with goodreason:

If a Republican wins the election, then if it’s not Reagan whowins it will be Anderson.

A Republican will win the election.

Yet they did not have reason to believe

63


If it’s not Reagan who wins, it will be Anderson.

I see what looks like a large fish writhing in a fisherman’s net aways off. I believe

If that creature is a fish, then if it has lungs, it’s a lungfish.

That, after all, is what one means by “lungfish.” Yet, even though Ibelieve the antecedent of this conditional, I do not conclude

If that creature has lungs, it’s a lungfish.

Lungfishes are rare, oddly shaped, and, to my knowledge, appearonly in fresh water. It is more likely that, even though it does notlook like one, the animal in the net is a porpoise.

Having learned that gold and silver were both once mined in hisregion, Uncle Otto has dug a mine in his backyard. Unfortunately,it is virtually certain that he will find neither gold nor silver, and itis entirely certain that he will find nothing else of value. There isample reason to believe

If Uncle Otto doesn’t find gold, then if he strikes it rich, itwill be by finding silver.

Uncle Otto won’t find gold.

Since, however, his chances of finding gold, though slim, are noslimmer than his chances of finding silver, there is no reason to sup-pose that

If Uncle Otto strikes it rich, it will be by finding silver.

64 VANN MCGEE


These examples show that modus ponens is not an entirely reliable ruleof inference. Sometimes the conclusion of an application of modus po-nens is something we do not believe and should not believe, eventhough the premises are propositions we believe very properly.4

Modus ponens is sometimes thought of not as a rule of inference butas a law of semantics, to wit, whenever ┌If φ then ψ┐ and φ are bothtrue, ψ is true as well. It is not at all obvious what we are to make of thislaw, since it is not evident what the truth conditions for the English con-ditional are or even whether it has truth conditions. Still it seems unlikelythat, even if we learned the truth conditions for the English conditional,the semantic version of modus ponens would be vindicated. Let us imag-ine, on the contrary, that some time in the future linguists will determinethe truth conditions for the English conditional and prove that modusponens is truth-preserving. Assuming that basic zoology will not havechanged, a future linguist who sees what looks like a large fish writhingin a fisherman’s net a ways off will believe, as I believed,

If that animal is a fish, then if it has lungs it’s a lungfish.That animal is a fish.

Suppose he also believes this:

It is true that, if that animal is a fish, then if it has lungs it’s alungfish.It is true that that animal is a fish.

Then he will be able to prove, using the well-established principle offuture semantics that modus ponens is truth-preserving:

It is true that, if that animal has lungs, it is a lungfish.

7 A Counterexample to Modus Ponens 65


He will not, however, believe

If that animal has lungs, it is a lungfish.

any more than I did. Thus our future linguist will be either in the awk-ward position of believing the premises of the argument without be-lieving that those premises are true, or else in the equally awkwardposition of not believing the conclusion of the argument even thoughhe does believe that that conclusion is true.5 Thus the only way that wecan hold on to the doctrine that modus ponens is truth-preserving willbe to accept an unexpected disparity between believing a propositionand believing that that proposition is true.

In an attempt to supply truth conditions where nature provides none,philosophers have settled upon material implication: Count ┌If φ thenψ┐ as true if either φ is false or ψ is true. Sometimes this is intended asa proposal for linguistic reform, a suggestion that, at least in our scien-tific discourse, we ought to use the “If-then” construction in a new way,treating it as the material conditional rather than the ordinary condi-tional. Our examples do not raise any difficulties for this proposal, sinceif we reinterpret them this way, our examples become arguments withtrue premises and true conclusions. Sometimes, however, material im-plication is proposed as an account of how we presently use the “If-then” construction. This is surely wrong. If we have seen the pollsshowing Reagan far ahead of Carter, who is far ahead of Anderson, wewill not for a moment suppose that

If Reagan doesn’t win, Anderson will.

is true, even though we will resign ourselves to the truth of

Reagan will win.

66 VANN MCGEE


Our counterexamples to modus ponens have a characteristic logicalform. Each has as a premise a conditional whose consequent is itself aconditional. In general, we assert, accept, or believe a conditional ofthe form ┌If φ, then if ψ then φ┐ whenever we are willing to assert,accept, or believe the conditional ┌If φ and ψ, then θ┐. It appears, fromlooking at examples, that the law of exportation.

┌If φ and ψ, then θ ┐ entails ┌If φ, then if ψ then θ ┐.

is a feature of English usage.6 If so, then our counterexamples to modusponens are not isolated curiosities but rather symptoms of a basic dif-ficulty. It is natural to suppose that the English indicative conditionalis intermediate in strength between strict implication and material im-plication. That is to say, whenever ψ is a logical consequence of φ, ┌Ifφ then ψ┐ will be true, and whenever ┌If φ then ψ┐ is true, either φwill be false or ψ true (and so modus ponens is truth-preserving). Itnow appears that we also want to require that the law of exportation bevalid. But there is no connective other than the material conditionalthat meets all these requirements.

Theorem. Suppose that we have a logical consequence relation ├ on alanguage whose connectives comprise the ordinary Booleanconnectives ‘∨’, ‘&’, ‘~’, ‘⊃’, and ‘≡’, as well as an additionalconditional ‘⇒’, satisfying the following conditions:

(Cons) ├, a relation between sets of sentences and sentences, isa consequence relation:If φ ∈ Γ, then Γ ├ φIf Γ ├ φ and Γ ⊆ Δ, then Δ ├ φ.If Δ ├ ψ for each ψ ∈ Γ and Γ ├ φ, then Δ ├ φ.

(Exp) The law of exportation for “⇒”:{┌φ & ψ ⇒ θ ┐ } ├ ┌φ ⇒ (ψ ⇒ θ )┐.



(MP) Modus ponens for both conditionals “⇒” and “⊃”:{┌φ ⇒ ψ┐,φ} ├ ψ{┌φ ⊃ ψ ┐,φ} ├ ψ

(StrImp) Strict implication is as strong or stronger than eitherconditional: If {φ}├ ψ, then Φ ├ ┌φ ⇒ ψ┐ and Φ├ ┌φ ⊃ ψ ┐ (where Φ is the empty set).

(Taut) Ordinary Boolean connectives behave normally: If φ is a tautology,7 then Φ ├ φ.8

Then the two conditionals “⇒” and “⊃” are logically indistinguish-able. More precisely, if φ and φ’ are alike except that ‘⇒’ and ‘⊃’ havebeen exchanged at some places, then {φ} ├ φ’ and {φ’} ├ φ.

The idea of the proof, which proceeds by induction on the complexityof φ, is contained in the proof that {┌ ψ ⊃ θ ┐} ├ ┌ ψ ⇒ θ ┐:9

(i) Φ ├ ┌ ((ψ ⊃ θ) & ψ) ⊃ θ ┐ by (Taut).(ii) {┌ ((ψ ⊃ θ) & ψ) ⊃ θ ┐; ┌ (ψ ⊃ θ) & ψ ┐} ├ θ by (MP) for ‘⊃’

(iii) {┌ (ψ ⊃ θ) & ψ┐} ├ θ from (i) and (ii) by (Cons)(iv) Φ ├ ┌((ψ ⊃ θ) & ψ ⇒ θ ┐ from (iii) by (StrImp) for ‘⇒’(v) {┌((ψ ⊃ θ) & ψ ) ⇒ θ ┐} ├ ┌ (ψ ⊃ θ) ⇒ ( ψ ⇒ θ) ┐ by (Exp)

(vi) {┌(ψ ⊃ θ) ⇒ ( ψ ⇒ θ) ┐; ┌ ψ ⊃ θ ┐ } ├ ┌ ψ ⇒ θ ┐

by (MP) for ‘⇒’(vii) {┌ ψ ⊃ θ┐} ├ ┌ ψ ⇒ θ ┐ from (iv), (v), and (vi) by (Cons)

The theorem points to a tension between modus ponens and the lawof exportation. According to the classical account, which does not rec-ognize any conditional other than the material, both are valid; but wewill not expect them both to come out valid on any nonclassical account.

We have explicit examples to show that the indicative conditionaldoes not satisfy modus ponens. It is not so easy to test whether the rule

68 VANN MCGEE


is valid for the subjunctive conditional, since we seldom use the sub-junctive conditional in situations in which we are confident that the an-tecedent is true. On the other hand, it is easy to find natural instances ofthe law of exportation that employ the subjunctive mood; for example,

If Juan hadn’t married Xochitl and Sylvia hadn’t run off to India,Juan and Sylvia would have become lovers.

entails

If Juan hadn’t married Xochitl, then if Sylvia hadn’t run off toIndia, Juan and Sylvia would have become lovers.

Multiplying such examples, we get good inductive evidence that thesubjunctive conditional satisfies the law of exportation. If this evidenceis correct, then no theory of the subjunctive conditional which deniesthe law of exportation will be entirely accurate. The most prominentlogical theory of the subjunctive conditional is Robert Stalnaker’s ac-count,10 according to which we test whether ┌φ ⇒ ψ ┐ is true in a pos-sible world ω by seeing whether ψ is true in the possible world mostsimilar to ω in which φ is true. Stalnaker’s system satisfies conditions(Cons), (MP), (StrImp), and (Taut), but it does not satisfy the law ofexportation. Thus we are led to suspect that Stalnaker’s analysis of thesubjunctive conditional is inaccurate.

Concrete examples confirm our suspicions. We would ordinarily say(at least in contexts in which we are interested in the election resultsrather than, say, how else the primaries might have turned out),

If Reagan hadn’t won the election and a Republican had won, itwould have been Anderson.



Appropriately, the Stalnaker semantics, under the natural comparitivesimilarity ordering among worlds, has this sentence come out true. Asthe law of exportation predicts, we also want to say,

If Reagan hadn’t won the election, then if a Republican had won,it would have been Anderson.

However, the possible world most similar to the actual world in whichReagan did not win the election will be a world in which Carter finishedfirst and Reagan second, with Anderson again a distant third, and so aworld in which “If a Republican had won it would have been Reagan”is true. Thus Stalnaker’s theory wrongly predicts that, in the actualworld,

If Reagan hadn’t won the election, then if a Republican had won,it would have been Reagan.

will be true. Thus, in this instance, the law of exportation is right andthe Stalnaker semantics is wrong.

Another example: Let us imagine that, contrary to all our expecta-tions, Uncle Otto finds a rich vein of gold, deeply buried in a distantcorner of his property. We still believe this:

If Uncle Otto hadn’t found gold but he had struck it rich, it wouldhave been by finding silver.

We also believe, as the law of exportation predicts,

If Uncle Otto hadn’t found gold, then if he had struck it rich, itwould have been by finding silver.

70 VANN MCGEE


What does the Stalnaker semantics say? The closest world to the actualworld in which Uncle Otto does not find gold—call it ω—will be aworld in which the deposit of gold is located just on the other side ofOtto’s property line, or perhaps a world in which Otto does not digquite deeply enough to reach the vein. The world closest to ω in whichOtto strikes it rich will be a world in which the gold is relocated backonto Otto’s property and Otto digs deeply enough to find the gold. Thusthe closest world to ω in which Uncle Otto strikes it rich will be a worldin which

Uncle Otto finds gold.

is true. Therefore, in ω,

If Uncle Otto had struck it rich, it would have been by findinggold

is true, and so, according to Stalnaker’s semantics,

If Uncle Otto hadn’t found gold, then if he had struck it rich, itwould have been by finding gold.

is true in the actual world. Once again, the law of exportation scores apoint against the Stalnaker semantics.

Our examples show us that an accurate logic for the English indica-tive conditional would have to restrict the rule of modus ponens some-how, and they suggest that the same would be true of an accurate logicof the subjunctive conditional. Nevertheless, all the familiar logics ofthe conditional countenance modus ponens without reservations. Howdo we account for this discrepancy? The simplest diagnosis is that we



have committed an error of overly hasty generalization. We encountera great many conditionals in daily life, and we have noticed that, whenwe accept a conditional and we accept its antecedent, we are prone toaccept the consequent as well. We have supposed that this pattern helduniversally, with no exceptions. However, the examples we looked atwere nearly always examples of simple conditionals, conditionals thatdid not themselves contain conditionals. Indeed there is every reasonto suppose that, restricted to such conditionals, modus ponens is unex-ceptionable. But when we turn our attention to compound conditionals,new phenomena appear, and patterns that established themselves in thesimple cases are disrupted.

The methodological moral to be drawn from this is that, when weformulate general laws of logic, we ought to exercise the same sort ofcaution we exercise when we make inductive generalizations in the em-pirical sciences. We must take care that the instances we look at in eval-uating a proposed generalization are diverse as well as numerous.

It is perhaps surprising that, in constructing a logical theory, onecomes upon the same pitfalls one encounters in the empirical sciences,since it is widely believed that logic is an a priori science. Upon reflec-tion, however, we see that there is no cause for perplexity. If one be-lieves that the correctness of a logically valid inference is recognizedby an a priori intuition, what one believes is this:

If ℜ is a valid rule of inference, then whenever R is an instance ofℜ, one can see by an a priori intuition that R is a correct inference.

In order to conclude that the general laws of logic can be establishedpurely by a priori reasoning, we would have to know somethingstronger, namely,

72 VANN MCGEE


If ℜ is a valid rule of inference, then one can see by an a prioriintuition that, whenever R is an instance of ℜ, R is a correctinference.

Our examples show that modus ponens is not strictly valid. They donothing to dissuade us from our entrenched belief that modus ponensis valid for simple conditionals. They suggest that the law of exporta-tion is valid for a wide range of cases, perhaps even valid universally.Beyond this, the examples give us no positive guidance toward con-structing a correct logic of conditionals. It may be that some entirelynew approach is needed, but it may also be that we can modify someexisting theory to take the examples into account.

It is not hard to modify the Stalnaker semantics so that it has theright logical features. Instead of the simple notion of truth in a world,we develop a notion of truth in a world under a set of hypotheses. Tobe simply true in a world is to be true in that world under the empty setof hypotheses. If there is no world accessible from ω in which all themembers of Γ are true, then every sentence is true in ω under the set ofhypotheses Γ. Otherwise we have the following: An atomic sentenceis true in ω under the set of hypotheses Γ iff it is true in the possibleworld most similar to ω in which all the members of Γ are true. A con-junction is true in a world under a given set of hypotheses if each of itsconjuncts is. A disjunction is true in a world under a set of hypothesesiff one or both disjuncts are. ┌~ φ ┐ is true in ω under the set of hy-potheses Γ iff φ is not true in ω under that set of hypotheses. Finally,┌φ ⇒ ψ┐ is true in ω under the set of hypotheses Γ iff ψ is true in ωunder the set of hypotheses Γ ∪ {φ}. Thus to evaluate whether ┌φ ⇒(ψ ⇒ θ )┐ is true under the set of hypotheses Γ, we add first φ andthen ψ to our set of hypotheses, and we see whether θ is true under



the augmented set of hypotheses Γ ∪ {φ, ψ}. This semantics gives alogic that is compact and decidable.

For each sentence constructed using this modified Stalnaker condi-tional, we can find a logically equivalent sentence that uses the originalStalnaker conditional. We use ‘⇒’ to stand for the modified Stalnakerconditional and ‘>’ to denote the connective Stalnaker originally de-scribed. We take the Boolean connectives to be ‘∨’, ‘&’, ‘~’, and a log-ically constant false sentence ‘⊥’. Define the operation * by:

φ * = φ if φ is an atomic sentence.‘⊥’* = ‘⊥’┌ (φ ∨ ψ) ┐* = ┌ (φ ∗ ∨ ψ*)┐┌ (φ & ψ) ┐* = ┌ (φ ∗ & ψ*)┐┌ ~ φ ┐ = ┌ ~ (φ ∗)┐┌ (φ ⇒ ψ) ┐* = ┌ (φ ∗ > ψ*)┐ if ψ is an atomic sentence or ‘⊥’┌ (φ ⇒ (ψ ∨ θ)) ┐* = ┌ ((φ ⇒ ψ)* ∨ (φ ⇒ θ)*)┐┌ (φ ⇒ (ψ & θ)) ┐* = ┌ ((φ ⇒ ψ)* & (φ ⇒ θ)*)┐┌ (φ ⇒ ∼ ψ) ┐* = ┌ ((φ ⇒ ⊥)*∨∼ ((φ ⇒ ψ)*))┐┌ (φ ⇒ (ψ ⇒ θ)) ┐* = ┌ ((φ & ψ ) ⇒ θ) ┐*

φ and φ * are logically equivalent.Another approach we might use would be to continue to use a for-

mal system in which modus ponens has unrestricted validity, and totake account of the invalidity of modus ponens in English by modify-ing our informal rules for translating English sentences into the formallanguage.11 Thus we do not translate an English sentence of the form┌If φ, then if ψ then θ┐; in the natural way, as a formula of the form┌(φ ⇒ (ψ ⇒ θ))┐; instead we translate it as ┌((φ & ψ) ⇒ θ)┐. Thusthe invalid English inference:

74 VANN MCGEE


If φ, then if ψ then θ.φ.

Therefore if ψ then θ.

is translated as the invalid formal inference:

(φ & ψ) ⇒ θ.φ.

Therefore ψ ⇒ θ.

It is sometimes a bit arbitrary whether to account for a feature ofEnglish usage within our formal system or to account for it at the in-formal level of translation lore. For example, we just discussed a wayof modifying the Stalnaker conditional so as to make the law of expor-tation generally valid. If we let Tr(φ) be the “natural” translation of anEnglish sentence φ into a formal language whose connectives are theBoolean connectives and ‘⇒’, we can equally well take the translationof φ to be Tr(φ) and use the modified Stalnaker semantics or take thetranslation of φ to be (Tr(φ))* and use the original Stalnaker semantics.

The selective use of unnatural translations is a powerful techniquefor improving the fit between the logic of the natural language and thelogic of a formal language. In fact, it is a little too powerful. One sus-pects that, if one is sly enough in giving translations, one can enablealmost any logic to survive almost any counterexample. What is neededis a systematic account of how to give the translations. In the absenceof such an account, the unnatural translations will seem like merely anad hoc device for evading counterexamples.

There is no guarantee that any approach will work. It may be that itis not possible to give a satisfactory logic of conditionals. This is not



to say that it is not possible to give a linguistic account of how we useconditionals, but only to say that such an account would not give riseto a tractable theory of logical consequence.

NOTES

1. I would like to thank Ernest Adams for his great help in preparing thispaper. He read the paper carefully and made a number of thoughtful andvaluable suggestions.

2. The corners, ‘┌’ and ‘┐’, are quasi-quotation marks. See Willard VanOrman Quine, Mathematical Logic (New York: Norton, 1940; Harper &Row, 1951), pp. 33–37.

3. Here I speak of inferring the sentence ψ from the sentences ┌If φ thenψ┐ and φ, and at other places I shall speak of inferring the proposition ψ fromthe propositions ┌If φ then ψ┐ and φ. It would be more precise, but also moretedious, to say that we infer the proposition expressed by the sentence ψ fromthe propositions expressed by the sentences ┌If φ then ψ┐ and φ.

4. There are, of course, familiar cases in which we see that an applicationof modus ponens leads us from premises we reasonably believe to a conclu-sion we find utterly incredible, and we respond by repudiating the premisesrather than accepting the conclusion. The present examples are not like this,since we do not renounce the premises.

5. The first horn of this dilemma would not be uncomfortable to someonelike Adams [The Logic of Conditionals (Boston: Reidel, 1975)] who doubtsthat conditionals are either true or false. By hypothesis, this is not the situa-tion of our future linguist.

6. It would appear that the law of importation, the converse of the law ofexportation, is also valid.

7. To see whether φ is a tautology, apply the following test: First replaceevery sub-formula of φ of the form ┌ψ ⇒ θ ┐ that is not itself contained insuch a subformula by a new sentential letter. Then apply the usual truth-tabletest.

8. We get an equivalent set of conditions by replacing (Exp) and (StrImp)for ‘⇒’ by the principle

76 VANN MCGEE


(Cond) If [Γ] ∪ {φ} ├ ψ, then Γ ├ ┌φ ⇒ ψ ┐.

This rule reflects the way we customarily prove conditionals: Add φ hy-pothetically to our body of theory. If we can prove ψ in the augmented theory,count ┌If φ then ψ ┐ as proved.

9. This conclusion already shows us that ‘⇒’ is not genuinely strongerthan the material conditional, as we would have hoped. Notice that to get itwe need only this very weak form of (StrImp):

If ψ is a tautological consequence of φ, then Φ ├ ┌φ ⇒ ψ ┐.

10. “A Theory of Conditionals,” in Nicholas Rescher, ed., Studies in Log-ical Theory. American Philosophical Quarterly supplementary monographseries (Oxford: Blackwell, 1968), pp. 98–112.

11. Barry Loewer, “Counterfactuals with Disjunctive Antecedents,” thisJournal, LXXIII, 16 (Sept. 16, 1976): 531–537, has proposed using this strat-egy for coping with a different difficulty with Stalnaker’s analysis.



8

Not a Counterexample to Modus Ponens

E. J. Lowe

In ‘A Counterexample to Modus Ponens’ (The Journal of PhilosophyLXXXII, 9, September 1985, pp. 462–71), Vann McGee presents thefollowing alleged counterexample to modus ponens (he also presentstwo others which, however, are constructed in essentially the sameway):

Opinion polls taken just before the 1980 election showed the Re-publican Ronald Reagan decisively ahead of the Democrat JimmyCarter, with the other Republican in the race, John Anderson, a dis-tant third. Those apprised of the polls results believed, with goodreason:

If a Republican wins the election, then if it’s not Reagan whowins it will be Anderson.

A Republican will win the election.


If it’s not Reagan who wins, it will be Anderson. (p. 462)

79


The example fails to serve McGee’s purpose because, I believe, it doesnot exhibit a genuine application of modus ponens—a rule which, inMcGee’s own words, ‘tells us that from an indicative conditional ┌Ifφ then ψ┐, together with its antecedent φ, one can infer ψ’ (ibid.).McGee clearly supposes that his example supplies premises of the form┌φ ⇒ (ψ⇒ θ)┐and φ, and a conclusion of the form ┌ ψ⇒ θ ┐, where‘⇒’ stands for a conditionship relation discernible in the English in-dicative conditional which, allegedly, is intermediate in strength be-tween strict implication and material implication (see ibid., p. 465). Myview, however, is that, while it may be allowed that many English in-dicative conditionals require to be interpreted as involving a condition-ship relation stronger than that of material implication (and which Ishall continue to represent by ‘⇒’), in fact the first premise of McGee’sexample has the form ┌φ ⇒ (ψ ⊃ θ)┐—so that the embedded condi-tional is a material conditional—whereas the conclusion is indeed ofthe form ┌ψ ⇒ θ┐. This being so, though, the failure of the conclusionto follow from the premises cannot be taken to impugn the validity ofmodus ponens, which is not exemplified by the inference in question;the only sort of fallacy involved in passing from these premises to thisconclusion is purely one of equivocation.

My point may be made as follows. Plausibly—or so I would urge—McGee’s first premise is just equivalent to

If a Republican wins the election, then either it will be Reaganwho wins or it will be Anderson.

Observe that from this and McGee’s second premise modus ponenswould have us infer

Either it will be Reagan who wins or it will be Anderson,

80 E. J. LOWE


which it is reasonable to believe in the circumstances described, be-cause it is reasonable to believe a disjunction when it is reasonable tobelieve one of its disjuncts. Of course, this disjunction is just equivalentto the material conditional whose antecedent is

Reagan will not win the election

and whose consequent is

Anderson will win it.

Clearly, however, it is not this material conditional that is expressed inMcGee’s example by the non-embedded sentence

If it’s not Reagan who wins, it will be Anderson

since we are invited to interpret this latter in a sense in which it ex-presses something unreasonable to believe in the circumstances de-scribed (unreasonable because of course in those circumstances it isreasonable to believe a conditional with the same antecedent but a con-trary consequent). We should appreciate, however, that this very formof words can be interpreted in either of two ways: either as sayingsomething of the form ┌ ψ ⇒ θ┐(which interpretation is demanded ofthe non-embedded occurrence of the sentence in McGee’s example),or as saying something of the form ┌ψ ⊃ θ┐—and that it is the latterinterpretation which is demanded where the sentence in question ap-pears embedded in McGee’s first premise. If it is wondered why thelatter interpretation is not readily available for the non-embedded con-ditional in McGee’s example, the answer is familiar and straightfor-ward enough: no conversational point is normally served by asserting

8 Not a Counterexample to Modus Ponens 81


something of the form ┌ψ ⊃ θ┐where ψ and θ are reasonably believedto be false, any more than it is by asserting something of the equivalentform ┌~ ψ ∨ θ┐. This indeed is why, although in the circumstances ofMcGee’s example it would be reasonable to believe the disjunction

Either it will be Reagan who wins the election or it will beAnderson,

it would not be conversationally appropriate to assert this.Perhaps it will be felt that I have not done enough to prove that

McGee’s first premise really is of the form ┌φ ⇒ (ψ ⊃ θ)┐. But it isenough that this is not a patently implausible interpretation, as I haveattempted to show by pointing to the plausible equivalence of thatpremise with a sentence of the form ┌φ ⇒ (~ ψ ∨ θ)┐. The burden ofproof lies rather with McGee to show that his first premise genuinelyis of the form ┌φ ⇒ (ψ ⇒θ)┐, since it is he who is relying on this as-sumption in order to challenge a deeply rooted principle of deductiveinference. As things stand, it is more reasonable to appeal to the validityof modus ponens to show that McGee has misinterpreted the form ofone of the sentences he invokes in his example. That is to say, a rea-sonable degree of logical conservatism entitles us to see in McGee’sexample not a breakdown of modus ponens but rather a demonstrationthat the English indicative conditional is sometimes interpretable as amaterial conditional and sometimes not—something which, as it hap-pens, I have argued for elsewhere on quite independent grounds (seemy ‘Indicative and Counterfactual Conditionals’, Analysis 39.3, June1979, pp. 139–41). And, more generally, the lesson to be drawn is thatlogicians should be wary of jettisoning long-established principles ofinference in response to the discovery of apparent counterexamples:

82 E. J. LOWE


almost always it is safer to conclude that one’s recalcitrant linguisticintuitions are erroneous or confused and to try to set them to rights bylaying bare distinctions to which one was blind.

One final point. Even if it is granted that McGee’s first premise ac-tually has the form ┌φ ⇒ (ψ ⊃ θ┐, it might still be asked whether thatvery sentence could also be interpreted as expressing something of theform ┌φ ⇒ (ψ ⇒ θ)┐and if so whether under such an interpretationit could be taken to express something reasonable to believe in the cir-cumstances described in McGee’s example. My answer is that I amfar from sure that the kind of conditionship relation represented by‘⇒’ genuinely admits of this sort of embedding, but that to the limitedextent that I can make sense of the suggestion that it does my reply tothe second part of the question is ‘No’. This is because the nearest Ican come to allowing the sentence in question the proposed interpre-tation is to suppose that it might be understood as expressing in an el-liptical way something that might more perspicuously be expressedby the following:

If a Republican wins the election, then it will have been true tosay that if it’s not Reagan who wins, it will be Anderson.

And this, to the extent that I can make sense of it, expresses somethingwhich I consider it would have been unreasonable for those in the cir-cumstances described in McGee’s example to believe. (As may begathered, one of the primary difficulties I see in the notion of embed-ding one ‘⇒’ conditional within another in the way McGee envisagesis the problem of accommodating their tenses, particularly where bothconditionals are future tense ones: for straightforwardly embeddingone such conditional within another as its consequent has the effect

8 Not a Counterexample to Modus Ponens 83


of transforming the tense of the embedded conditional from the futureto the future future, so that the embedded conditional is not in factequivalent to a non-embedded conditional of the same form—an effectwhich can only be overcome, and even then not entirely satisfactorily,by using the future perfect tense in the embedding conditional. Butthese are problems I cannot examine in detail here.)

84 E. J. LOWE


9

Assumptions and the Supposed

Counterexamples to Modus Ponens

D. E. Over

Is modus ponens a valid form of inference for the ordinary conditional?In a recent article Vann McGee has surprisingly argued ‘no’ on the basisof some very interesting examples. (See ‘A Counterexample to ModusPonens’ in The Journal of Philosophy, September 1985, pp. 462–71.) Idisagree with McGee’s conclusion about modus ponens, but think thathis examples should be used to make some important points about as-sumptions, beliefs, ordinary conditionals, and valid inference. Thesepoints will show that no real counterexamples to modus ponens havebeen given.

McGee presents a number of very closely related examples, and hedescribes (on p. 462) the first of these, which is also the most attractive,in the following way. (The numbering (l) – (3) has been added by me.)

Opinion polls taken just before the 1980 election showed the Re-publican Ronald Reagan decisively ahead of the Democrat JimmyCarter, with the other Republican in the race, John Anderson, a dis-tant third. Those apprised on the poll results believed, with goodreason:

85


(1) If a Republican wins the election, then if it’s not Rea-gan who wins it will be Anderson.

(2) A Republican will win.


(3) If it’s not Reagan who wins, it will be Anderson.

McGee’s conclusion in the light of examples of this type is thatmodus ponens is ‘not strictly valid’ (p. 462), and he expresses this pointin the following way (p. 463):

Sometimes the conclusion of an application of modus ponens issomething we do not believe and should not believe, even thoughthe premisses are propositions we believe very properly.

The first important point to notice is that McGee speaks of what webelieve in the above quotation, and not of what we assume. In fact, henever speaks of modus ponens as the rule which allows us to infer aconclusion ψ from assumptions of the form φ and φ ⇒ ψ (or from as-sumptions these forms depend on), and yet this is how the rule is stan-dardly stated in systems of natural deduction.1 Actually (3) does validlyfollow given that we assume (1) and (2). Having a belief is not at allthe same thing as making an assumption, and one respect in which theydiffer is that a belief may be suspended or set aside when an assumptionby its very nature cannot be.

Suppose you believe that Reagan will win the election, and I ask youto consider what will happen if he does not win. You have no difficulty—you suspend or set to one side (in very informal terms) your belief aboutReagan and certain other related beliefs, and then try to see what plau-

86 D. E. OVER


sibly follows. But suppose now I ask you what will happen if Reagandoes not win the election on the assumption that he will win it. This ispuzzling because I am asking you, in effect, what follows from a con-tradiction. You do not immediately suspend or give up the assumptionthat Reagan will win the election, in order to decide what will happenif he does not win it, for what in that case would be the point of makingthe assumption? An assumption is a proposition we hang on to in cir-cumstances like this, and to make the assumption is to agree for a timeto hang on to the proposition.

Let us return to (3) above, and imagine that we are presented withthis conditional on its own and asked whether we should believe it. Toanswer this question, we suppose that the antecedent of (3) is correctand, to avoid inconsistency, suspend some of our present beliefs, in-cluding, of course, our belief that Reagan will win the election. But bysuspending that belief, we see that we no longer have any reason to be-lieve (2) above, that a Republican will win (since we have presumablybased this belief on our belief that Reagan will win). And as a result ofthese changes and what we know about the polls, we see that we shouldnot believe (3), or more accurately the proposition which (3) expressesin this context.

The matter is very different in a context in which we infer (3) fromthe assumptions (1) and (2) by modus ponens. Here we cannot ‘sus-pend’ (2)—in this context (2) is an assumption, and the rule modus po-nens does not permit us to discharge its assumptions (modus ponens isnot like conditional introduction in this respect). And so (3) does followgiven the assumption (1) and (2), or more accurately the propositionhere expressed by (3) does follow from the propositions expressed bythe assumptions (1) and (2).

There are in addition serious arguments against any attempt likeMcGee’s to define validity in terms of justified belief. Of course, there

9 Assumptions and Counterexamples to Modus Ponens 87


is the well known point that the premisses of a valid argument may in-dividually have a higher probability than its conclusion, and somewould argue for the analogous point about justification.2 Although theprobability of φ may be relatively high and also the probability of ψ,the probability of φ & ψ may be relatively low. Admittedly, the proba-bility of the conjunction of the premisses cannot be higher than that ofthe conclusion in a valid inference, but some epistemologists have ar-gued that degree of justification is not similarly closed under knownvalid inference. They claim that one may not be justified in believinga proposition ψ, even though one is justified in believing the singleproposition φ and knows that ψ may be validly inferred from φ.3 Thisclaim of these epistemologists should be questioned, I think, but shouldnot be ruled out as trivially false by the very definition of validity. Fi-nally, how would validity be defined for inferences in which assump-tions were discharged? We would not want to be faced withMcGee-type ‘counterexamples’ to disjunction elimination and existen-tial elimination because assumptions had been discharged, leaving con-clusions apparently unsupported by justified beliefs alone.

All the points made so far here are relevant to a much wider rangeof examples than those discussed by McGee, as illustrated by thisinference:

(4) If the polls are right, then Reagan must win;(5) The polls are right;(6) Reagan must win.

Suppose we ask whether we should believe (6) above when wemerely have good reason to believe, but do not assume, (4) and (5).Surely (6) may strike us as too strong a statement to be worthy of beliefin this kind of case. If (5) expresses only a reasonable belief, and not

88 D. E. OVER


an assumption, then we may not hold it constant in the set of possiblesituations we use to determine the acceptability of what is here ex-pressed by (6). To suspend our belief in (5) is to be prepared to considerpossibilities in which Reagan does not win, and so these possibilitiesdetermine what is expressed by (6) and the un-acceptability of thisproposition in this case. But if we go to a context in which we take (4)and (5) as assumptions, then we see that the relevant set of possible sit-uations for judging the new use of (6) cannot have members in whicheither (4) or (5) is false. To assume (4) and (5) is to exclude such pos-sibilities, and thus the ‘narrower’ proposition which (6) expresses inthis context is true.

Some logicians would argue that there is a scope ambiguity in(4), and some of these would claim that (4) is only true (in the gen-eral circumstances described above) when it is given the logical form□ (φ ⇒ ψ). With (4) in this form, of course, (4) – (6) would no longerappear to be an instance of modus ponens, and one would not need toask whether (4) and (5) were assumptions. But this way of dealing with(4) – (6) as an apparent counterexample to modus ponens does not seemto be open to McGee, who appears to be committed to a hard line ongiving logical forms to sentences in ordinary English. Towards the endof his paper he considers the claim that the logical form of (1) shouldbe translated as (φ & ψ) ⇒ θ in a formal logic for the conditional.Clearly, if we only believed (1) in this form, then McGee could notargue that (l) – (3) is a counterexample to modus ponens. But he chargesthat it is unnatural to translate (1) in this way, and seems to take a hardline on translations in general in the following passage (p. 471):

The selective use of unnatural translations is a powerful techniquefor improving the fit between the logic of natural language andthe logic of a formal language. In fact, it is a little too powerful.



One suspects that, if only one is sly enough in giving translations,one can enable almost any logic to survive almost any counterex-ample. What is needed is a systematic account of how to givetranslations. In the absence of such an account, the unnaturaltranslations will seem likely merely an ad hoc device for evadingcounterexamples.

McGee does have a good point, to some extent, about the practiceof some logicians in trying to give such translations, but he himself isin serious trouble. His account of validity in terms of justified beliefand his strong comments on translation imply the very implausible re-sult that there are many types of quite straightforward counterexamplesto modus ponens. And we can bring out even more clearly what iswrong with these supposed counterexamples by considering the fol-lowing modification of (l) – (3):

(7) If a Republican wins, then if he is not Reagan he will beAnderson;

(8) A Republican will win;(9) If he is not Reagan, he will be Anderson.

The antecedent of (7) restricts the possibilities for the interpretationof the pronoun in its consequent. The second assumption (8) does thesame job for the conclusion (9), and it would be a transparent mistaketo try to interpret ‘he’ in some other way, in an attempt to show that(7) – (9) is invalid. McGee would make a mistake of this type if hethought of (8) as a relatively long-lasting mental state of justified beliefoutside of the context of this inference. He would not then see (8) asan assumption in an inference, determining in that context which propo-sition is expressed by (9).

90 D. E. OVER


An inference should be defined in terms of a relationship betweenassumptions and a conclusion, as is standard in logic. We should re-member that the assumptions can restrict the relevant set of possibilitiesand so affect the propositions expressed under them, just as the an-tecedents can affect the propositions expressed by the consequents ofconditionals. We must therefore be careful about the propositions ex-pressed in inferences, particularly ones containing conditionals, if wewish to question their validity.4 This is as true for (l) – (3) and (4) – (6)as it is for (7) – (9), and the point also applies to inferences of otherforms, such as modus tollens, which can appear to have McGee-type‘counterexamples’. And when we use any of these valid inferences toextend our justified beliefs, we must be careful about which proposi-tions we are to believe. Once these steps are taken, McGee-type ‘coun-terexamples’ to valid inference will disappear. Validity can be given aproper semantic definition in terms of propositions and truth preserva-tion, and on that firm basis epistemology can legitimately ask the ques-tion of how justified belief is related to validity.5

NOTES

1. As McGee himself admits in a footnote (p. 462), modus ponens shouldreally be seen as an inference involving propositions expressed by sentencesof certain forms—a point relevant to what follows.

2. This criticism of McGee is stressed by Walter Sinnott-Armstrong,James Moor, and Robert Fogelin, ‘A Defence of Modus Ponens,’ The Jour-nal of Philosophy, May 1986, pp. 296–300.

3. The influential source of this view seems to be Fred Dretske, ‘Epis-temic Operators,’ The Journal of Philosophy, 1970, pp. 1007–23. It is closelyrelated to the claim that knowledge is not closed under known logical im-plication—see also Robert Nozick, Philosophical Explanations (HarvardUniversity Press, 1981), chap. 3.

4. This point can be used to support those who claim that (3) should notbe interpreted in the same way as the consequent of (1). See, for example,



E. J. Lowe., ‘Not a Counterexample to Modus Ponens,’ Analysis 47.1, Jan-uary 1987, pp. 44–7.

5. For helpful discussion, I should like to thank Simon Blackburn,Dorothy Edgington, Hugh Mellor, Richard Spencer-Smith, and especiallyWilfred Hodges and E. J. Lowe. Very special thanks are due to Kit Fine, whomust be ultimately responsible for any good ideas in the paper.

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QUESTIONS

III. LOGIC AND INFERENCE

1. Following McGee, formulate your own counterexample to modusponens.

2. Why does Over hold that deductive inference may not provide epis-temic grounds for extending our beliefs?

3. According to Lowe, what is it to be a logical libertine?

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IV

LOGIC AND FREEDOM

According to the law of excluded middle, all statements are ei-ther true, or if not true, then false. But consider the statement“Tomorrow you will call the White House.” If the statement istrue, then you are bound to make the call. If the statement isfalse, you are bound not to make the call. In either case only onepossible course of action is open to you. So you are not free withregard to calling the White House tomorrow. And this argumentapplies to every one of your future actions. Thus you never actfreely. Is this argument sound?

Gilbert Ryle thinks not, asserting that “events themselvescannot be made necessary by truths.” Richard Taylor, however,argues that six presuppositions (including the law of excludedmiddle) widely accepted by contemporary philosophers implythe fatalistic conclusion that we have no more control over futureevents than we have now over past ones. Taylor and Steven M.Cahn offer a different argument leading to a fatalistic conclusion,claiming that we can only render true statements that are alreadytrue, and can only render false statements that are already false,and that even these conceptions make dubious sense.


10

‘It Was To Be’

Gilbert Ryle

I want now to launch out without more ado into the full presentationand discussion of a concrete dilemma. It is a dilemma which, I expect,has occasionally bothered all of us, though, in its simplest form, notvery often or for very long at a time. But it is intertwined with two otherdilemmas, both of which probably have seriously worried nearly all ofus. In its pure form it has not been seriously canvassed by any importantWestern philosopher, though the Stoics drew on it at certain points. Itwas, however, an ingredient in discussions of the theological doctrineof Predestination and I suspect that it has exerted a surreptitious influ-ence on some of the champions and opponents of Determinism.

At a certain moment yesterday evening I coughed and at a certainmoment yesterday evening I went to bed. It was therefore true on Sat-urday that on Sunday I would cough at the one moment and go to bedat the other. Indeed, it was true a thousand years ago that at certain mo-ments on a certain Sunday a thousand years later I should cough andgo to bed. But if it was true beforehand—forever beforehand—that Iwas to cough and go to bed at those two moments on Sunday, 25 Jan-uary 1953, then it was impossible for me not to do so. There would bea contradiction in the joint assertion that it was true that I would dosomething at a certain time and that I did not do it. This argument is

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perfectly general. Whatever anyone ever does, whatever happens any-where to anything, could not not be done or happen, if it was true be-forehand that it was going to be done or was going to happen. Soeverything, including everything that we do, has been definitivelybooked from any earlier date you like to choose. Whatever is, was tobe. So nothing that does occur could have been helped and nothing thathas not actually been done could possibly have been done.

This point, that for whatever takes place it was antecedently true thatit was going to take place, is sometimes picturesquely expressed by say-ing that the Book of Destiny has been written up in full from the begin-ning of time. A thing’s actually taking place is, so to speak, merely theturning up of a passage that has for all time been written. This picturehas led some fatalists to suppose that God, if there is one, or, we our-selves, if suitably favoured, may have access to this book and readahead. But this is a fanciful embellishment upon what in itself is a severeand seemingly rigorous argument. We may call it ‘the fatalist argument.’

Now the conclusion of this argument from antecedent truth, namelythat nothing can be helped, goes directly counter to the piece of com-mon knowledge that some things are our own fault, that some threat-ening disasters can be foreseen and averted, and that there is plenty ofroom for precautions, planning and weighing alternatives. Even whenwe say nowadays of someone that he is born to be hanged or not bornto be drowned, we say it as a humorous archaism. We really think thatit depends very much on himself whether he is hanged or not, and thathis chances of drowning are greater if he refuses to learn to swim. Yeteven we are not altogether proof against the fatalist view of things. Ina battle I may well come to the half-belief that either there exists some-where behind the enemy lines a bullet with my name on it, or theredoes not, so that taking cover is either of no avail or else unnecessary.In card-games and at the roulette-table it is easy to subside into the

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frame of mind of fancying that our fortunes are in some way pre-arranged, well though we know that it is silly to fancy this.

But how can we deny that whatever happens was booked to happenfrom all eternity? What is wrong with the argument from antecedenttruth to the inevitability of what the antecedent truths are antecedentlytrue about? For it certainly is logically impossible for a prophecy to betrue and yet the event prophesied not to come about.

We should notice first of all that the premiss of the argument doesnot require that anyone, even God, knows any of these antecedent truths,or to put it picturesquely, that the Book of Destiny has been written byanybody or could be perused by anybody. This is just what distinguishesthe pure fatalist argument from the mixed theological argument for pre-destination. This latter argument does turn on the supposition that Godat least has foreknowledge of what is to take place, and perhaps alsopreordains it. But the pure fatalist argument turns only on the principlethat it was true that a given thing would happen, before it did happen,i.e. that what is, was to be; not that it was known by anyone that it wasto be. Yet even when we try hard to bear this point in mind, it is veryeasy inadvertently to reinterpret this initial principle into the supposi-tion that before the thing happened it was known by someone that itwas booked to happen. For there is something intolerably vacuous inthe idea of the eternal but unsupported pre-existence of truths in the fu-ture tense. When we say ‘a thousand years ago it was true that I shouldnow be saying what I am,’ it is so difficult to give any body to this ‘it’of which we say that it was then true, that we unwittingly fill it out withthe familiar body of an expectation which someone once entertained,or of a piece of foreknowledge which someone once possessed. Yet todo this is to convert a principle which was worrying because, in a way,totally truistic, into a supposition which is unworrying because quasi-historical, entirely without evidence and most likely just false.

10 ‘It Was To Be’ 99


Very often, though certainly not always, when we say ‘it was truethat . . .’ or ‘it is false that . . .’ we are commenting on some actual pro-nouncement made or opinion held by some identifiable person. Some-times we are commenting in a more general way on a thing which somepeople, unidentified and perhaps unidentifiable, have believed or nowbelieve. We can comment on the belief in the Evil Eye without beingable to name anyone who held it; we know that plenty of people didhold it. Thus we can say ‘it was true’ or ‘it is false’ in passing verdictsupon the pronouncements both of named and of nameless authors. Butin the premiss of the fatalist argument, namely that it was true beforesomething happened that it would happen, there is no implication ofanyone, named or unnamed, having made that prediction.

There remains a third thing that might be meant by ‘it was true athousand years ago that a thousand years later these things would bebeing said in this place,’ namely that if anybody had made a predictionto this effect, though doubtless nobody did, he would have been right.It is not a case of an actual prediction having come true but of a con-ceivable prediction having come true. The event has not made an actualprophecy come true. It has made a might-have-been prophecy cometrue.

Or can we say even this? A target can be hit by an actual bullet, butcan it be hit by a might-have-been bullet? Or should we rather say onlythat it could have been hit by a might-have-been bullet? The historical-sounding phrases ‘came true,’ ‘made true’ and ‘was fulfilled’ apply wellenough to predictions actually made, but there is a detectable twist,which may be an illegitimate twist, in saying that a might-have-beenprediction did come true or was made true by the event. If an unbackedhorse wins a race, we can say that it would have won money for itsbackers, if only there had been any. But we cannot say that it did winmoney for its backers, if only there had been any. There is no answer

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to the question ‘How much money did it win for them?’ Correspond-ingly, we cannot with a clear conscience say of an event that it has ful-filled the predictions of it which could have been made, but only thatit would have fulfilled any predictions of it which might have beenmade. There is no answer to the question ‘Within what limits of preci-sion were these might-have-been predictions correct about the time andthe loudness of my cough?’

Let us consider the notions of truth and falsity. In characterizingsomebody’s statement, for example a statement in the future tense, astrue or as false, we usually though not always, mean to convey rathermore than that what was forecast did or did not take place. There issomething of a slur in ‘false’ and something honorific in ‘true,’ somesuggestion of the insincerity or sincerity of its author, or some sugges-tion of his rashness or cautiousness as an investigator. This is broughtout by our reluctance to characterize either as true or as false pure andavowed guesses. If you make a guess at the winner of the race, it willturn out right or wrong, correct or incorrect, but hardly true or false.These epithets are inappropriate to avowed guesses, since the one epi-thet pays an extra tribute, the other conveys an extra adverse criticismof the maker of the guess, neither of which can he merit. In guessingthere is no place for sincerity or insincerity, or for caution or rashnessin investigation. To make a guess is not to give an assurance and it isnot to declare the result of an investigation. Guessers are neither reliablenor unreliable.

Doubtless we sometimes use ‘true’ without intending any connota-tion of trustworthiness and, much less often, ‘false’ without any con-notation of trust misplaced. But, for safety’s sake, let us reword thefatalist argument in terms of these thinner words, ‘correct’ and ‘incor-rect.’ It would now run as follows. For any event that takes place, anantecedent guess, if anyone had made one, that it was going to take



place, would have been correct, and an antecedent guess to the contrary,if anyone had made it, would have been incorrect. This formulation al-ready sounds less alarming than the original formulation. The word‘guess’ cuts out the covert threat of foreknowledge, or of there beingbudgets of antecedent forecasts, all meriting confidence before theevent. What, now, of the notion of guesses in the future tense beingcorrect or incorrect?

Antecedently to the running of most horse-races, some people guessthat one horse will win, some that another will. Very often every horsehas its backers. If, then, the race is run and won, then some of the back-ers will have guessed correctly and the rest will have guessed incor-rectly. To say that someone’s guess that Eclipse would win was correctis to say no more than that he guessed that Eclipse would win andEclipse did win. But can we say in retrospect that his guess, which hemade before the race, was already correct before the race? He madethe correct guess two days ago, but was his guess correct during thosetwo days? It certainly was not incorrect during those two days, but itdoes not follow, though it might seem to follow, that it was correct dur-ing those two days. Perhaps we feel unsure which we ought to say,whether that his guess was correct during those two days, though noone could know it to be so, or only that, as it turned out, it was duringthose two days going to prove correct, i.e. that the victory which did,in the event, make it correct had not yet happened. A prophecy is notfulfilled until the event forecast has happened. Just here is where ‘cor-rect’ resembles ‘fulfilled’ and differs importantly from ‘true.’ The hon-orific connotations of ‘true’ can certainly attach to a person’s forecastsfrom the moment at which they are made, so that if these forecasts turnout incorrect, while we withdraw the word ‘true,’ we do not necessarilywithdraw the testimonials which it carried. The establishment of incor-rectness certainly cancels ‘true’ but not, as a rule, so fiercely as to in-cline us to say ‘false.’

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The words ‘true’ and ‘false’ and the words ‘correct’ and ‘incorrect’are adjectives, and this grammatical fact tempts us to suppose that true-ness and falseness, correctness and incorrectness, and even, perhaps,fulfilledness and unfulfilledness must be qualities or properties residentin the propositions which they characterize. As sugar is sweet and whitefrom the moment it comes into existence to the moment when it goesout of existence, so we are tempted to infer, by parity of reasoning, thatthe trueness or correctness of predictions and guesses must be featuresor properties which belong all the time to their possessors, whether wecan detect their presence in them or not. But if we consider that ‘de-ceased,’ ‘lamented’ and ‘extinct’ are also adjectives, and yet certainlydo not apply to people or mastodons while they exist, but only afterthey have ceased to exist, we may feel more cordial towards the ideathat ‘correct’ is in a partly similar way a merely obituary and valedic-tory epithet, as ‘fulfilled’ more patently is. It is more like a verdict thana description. So when I tell you that if anyone had guessed that Eclipsewould win today’s race his guess would have turned out correct, I giveyou no more information about the past than is given by the eveningnewspaper which tells you that Eclipse won the race.

I want now to turn to the fatalist conclusion, namely that since what-ever is was to be, therefore nothing can be helped. The argument seemsto compel us to say that since the antecedent truth requires the event ofwhich it is the true forecast, therefore this event is in some disastrousway fettered to or driven by or bequeathed by that antecedent truth—as if my coughing last night was made or obliged to occur by the an-tecedent truth that it was going to occur, perhaps in something like theway in which gunfire makes the windows rattle a moment or two afterthe discharge. What sort of necessity would this be?

To bring this out let us by way of contrast suppose that someone pro-duced the strictly parallel argument, that for everything that happens,it is true for ever afterwards that it happened.



I coughed last night, so it is true today and will be true a thousandyears hence that I coughed last night. But these posterior truths in thepast tense, could not be true without my having coughed. Therefore mycoughing was necessitated or obliged to have happened by the truth ofthese posterior chronicles of it. Clearly something which disturbed usin the original form of the argument is missing in this new form. Wecheerfully grant that the occurrence of an event involves and is involvedby the truth of subsequent records, actual or conceivable, to the effectthat it occurred. For it does not even seem to render the occurrence aproduct or effect of these truths about it. On the contrary, in this casewe are quite clear that it is the occurrence which makes the posteriortruths about it true, not the posterior truths which make the occurrenceoccur. These posterior truths are shadows cast by the events, not theevents shadows cast by these truths about them, since these belong tothe posterity, not to the ancestry of the events.

Why does the fact that a posterior truth about an occurrence requiresthat occurrence not worry us in the way in which the fact that an ante-rior truth about an occurrence requires that occurrence does worry us?Why does the slogan ‘Whatever is, always was to be’ seem to implythat nothing can be helped, where the obverse slogan ‘Whatever is, willalways have been’ does not seem to imply this? We are not exercisedby the notorious fact that when the horse has already escaped it is toolate to shut the stable door. We are sometimes exercised by the ideathat as the horse is either going to escape or not going to escape, to shutthe stable door beforehand is either unavailing or unnecessary. A largepart of the reason is that in thinking of a predecessor making its suc-cessor necessary we unwittingly assimilate the necessitation to causalnecessitation. Gunfire makes windows rattle a few seconds later, butrattling windows do not make gunfire happen a few seconds earlier,even though they may be perfect evidence that gunfire did happen a

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few seconds earlier. We slide, that is, into thinking of the anterior truthsas causes of the happenings about which they were true, where the merematter of their relative dates saves us from thinking of happenings asthe effects of those truths about them which are posterior to them.Events cannot be the effects of their successors, any more than we canbe the offspring of our posterity.

So let us look more suspiciously at the notions of necessitating, mak-ing, obliging, requiring and involving on which the argument turns.How is the notion of requiring or involving that we have been workingwith related to the notion of causing?

It is quite true that a backer cannot guess correctly that Eclipse willwin without Eclipse winning and still it is quite false that his guessingmade or caused Eclipse to win. To say that his guess that Eclipse wouldwin was correct does logically involve or require that Eclipse won. Toassert the one and deny the other would be to contradict oneself. To saythat the backer guessed correctly is just to say that the horse which heguessed would win, did win. The one assertion cannot be true withoutthe other assertion being true. But in this way in which one truth mayrequire or involve another truth, an event cannot be one of the implica-tions of a truth. Events can be effects, but they cannot be implications.Truths can be consequences of other truths, but they cannot be causesof effects or effects of causes.

In much the same way, the truth that someone revoked involves thetruth that he had in his hand at least one card of the suit led. But he wasnot forced or coerced into having a card of that suit in his hand by thefact that he revoked. He could not both have revoked and not had acard of that suit in his hand, but this ‘could not’ does not connote anykind of duress. A proposition can imply another proposition, but it can-not thrust a card into a player’s hand. The questions, what makes thingshappen, what prevents them from happening, and whether we can help



them or not, are entirely unaffected by the logical truism that a state-ment to the effect that something happens, is correct if and only if ithappens. Lots of things could have prevented Eclipse from winning therace; lots of other things could have made his lead a longer one. Butone thing had no influence on the race at all, namely the fact that if any-one guessed that he would win, he guessed correctly.

We are now in a position to separate out one unquestionable andvery dull true proposition from another exciting but entirely falseproposition, both of which seem to be conveyed by the slogan ‘Whatis, always was to be.’ It is an unquestionable and very dull truth thatfor anything that happens, if anyone had at any previous time made theguess that it would happen, his guess would have turned out correct.The twin facts that the event could not take place without such a guessturning out correct and that such a guess could not turn out correct with-out the event taking place tell us nothing whatsoever about how theevent was caused, whether it could have been prevented, or evenwhether it could have been predicted with certainty or probability fromwhat had happened before. The menacing statement that what is wasto be, construed in one way, tells us only the trite truth that if it is trueto say (a) that something happened, then it is also true to say (b) thatthat original statement (a) is true, no matter when this latter comment(b) on the former statement (a) may be made.

The exciting but false proposition that the slogan seems to forceupon us is that whatever happens is inevitable or doomed, and, whatmakes it sound even worse, logically inevitable or logically doomed—somewhat as it is logically inevitable that the immediate successor ofany even number is an odd number. So what does ‘inevitable’ mean?An avalanche may be, for all practical purposes, unavoidable. A moun-taineer in the direct path of an avalanche can himself do nothing to stopthe avalanche or get himself out of its way, though a providential earth-

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quake might conceivably divert the avalanche or a helicopter mightconceivably lift him out of danger. His position is much worse, but onlymuch worse, than that of a cyclist half a mile ahead of a lumberingsteam-roller. It is extremely unlikely that the steam-roller will catch upwith him at all, and even if it does so it is extremely likely that its driverwill halt or that the cyclist himself will move off in good time. Butthese differences between the plights of the mountaineer and the cyclistare differences of degree only. The avalanche is practically unavoid-able, but it is not logically inevitable. Only conclusions can be logicallyinevitable, given the premisses, and an avalanche is not a conclusion.The fatalist doctrine, by contrast, is that everything is absolutely andlogically inevitable in a way in which the avalanche is not absolutelyor logically inevitable; that we are all absolutely and logically power-less where even the hapless mountaineer is only in a desperate plightand the cyclist is in no real danger at all; that everything is fettered bythe Law of Contradiction to taking the course it does take, as odd num-bers are bound to succeed even numbers. What sort of fetters are thesepurely logical fetters?

Certainly there are infinitely many cases of one truth making neces-sary the truth of another proposition. The truth that today is Mondaymakes necessary the truth of the proposition that tomorrow is Tuesday.It cannot be Monday today without tomorrow being Tuesday. A personwho said ‘It is Monday today but not Tuesday tomorrow’ would be tak-ing away with his left hand what he was giving with his right hand. Butin the way in which some truths carry other truths with them or makethem necessary, events themselves cannot be made necessary by truths.Things and events may be the topics of premisses and conclusions, butthey cannot themselves be premisses or conclusions. You may prefacea statement by the word ‘therefore,’ but you cannot pin either a ‘there-fore’ or a ‘perhaps not’ on to a person or an avalanche. It is a partial



parallel to say that while a sentence may contain or may be without asplit infinitive, a road accident cannot either contain or lack a split in-finitive, even though it is what a lot of sentences, with or without splitinfinitives in them, are about. It is true that an avalanche may be prac-tically inescapable and the conclusion of an argument may be logicallyinescapable, but the avalanche has not got—nor does it lack—the in-escapability of the conclusion of an argument. The fatalist theory triesto endue happenings with the inescapability of the conclusions of validarguments. Our familiarity with the practical inescapability of somethings, like some avalanches, helps us to yield to the view that reallyeverything that happens is inescapable, only not now in the way inwhich some avalanches are inescapable and others not, but in the wayin which logical consequences are inescapable, given their premisses.The fatalist has tried to characterize happenings by predicates whichare proper only to conclusions of arguments. He tried to flag my coughwith a Q.E.D.

Before standing back to draw some morals from this dilemma be-tween whatever is was to be and some things which have happenedcould have been averted, I want briefly to discuss one further pointwhich may be of only domestic interest to professional philosophers.If a city-engineer has constructed a roundabout where there had beendangerous cross-roads, he may properly claim to have reduced the num-ber of accidents. He may say that lots of accidents that would otherwisehave occurred have been prevented by his piece of road improvement.But suppose we now ask him to give us a list of the particular accidentswhich he has averted. He can do nothing but laugh at us. If an accidenthas not happened, there is no ‘it’ to put down on a list of ‘accidentsprevented.’ He can say that accidents of such and such kinds whichused to be frequent are now rare. But he cannot say ‘Yesterday’s colli-sion at midday between this fire-engine and that milk-float at this corner

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was, fortunately, averted.’ There was no such collision, so he cannotsay ‘This collision was averted.’ To generalize this, we can never pointto or name a particular happening and say of it ‘This happening wasaverted,’ and this logical truism seems to commit us to saying ‘No hap-penings can be averted’ and consequently ‘it’s no good trying to ensureor prevent anything happening.’ So when we try to say that some thingsthat happen could have been prevented; that some drownings, for ex-ample, would not have occurred had their victims learned to swim, weseem to be in a queer logical fix. We can say that a particular personwould not have drowned had he been able to swim. But we cannot quitesay that his lamented drowning would have been averted by swimming-lessons. For had he taken those lessons, he would not have drowned,and then we would not have had for a topic of discussion just thatlamented drowning of which we want to say that it would have beenprevented. We are left bereft of any ‘it’ at all. Averted fatalities are notfatalities. In short, we cannot, in logic, say of any designated fatalitythat it was averted—and this sounds like saying that it is logically im-possible to avert any fatalities.

The situation is parallel to the following. If my parents had nevermet, I should not have been born, and had Napoleon known somethings that he did not know the Battle of Waterloo would not have beenfought. So we want to say that certain contingencies would have pre-vented me from being born and the Battle of Waterloo from beingfought. But then there would have been no Gilbert Ryle and no Battleof Waterloo for historians to describe as not having been born and asnot having been fought. What does not exist or happen cannot benamed, individually indicated or put on a list, and cannot therefore becharacterized as having been prevented from existing or happening. Sothough we are right to say that some sorts of accidents can be pre-vented, we cannot put this by saying that this designated accident might



have been prevented from occurring—not because it was of an unpre-ventable sort, but because neither ‘preventable’ nor ‘unpreventable’ canbe epithets of designated occurrences, any more than ‘exists’ or ‘doesnot exist’ can be predicated of designated things or persons. As ‘unborn’cannot without absurdity be an epithet of a named person, so ‘born’cannot without a queerly penetrating sort of redundancy be an epithetof him either. The question ‘Were you born or not?’ is, unless specialinsurance-policies are taken out, an unaskable question. Who could beasked it? Nor could one ask whether the Battle of Waterloo was foughtor unfought. That it was fought goes with our having an it to talk aboutat all. There could not be a list of unfought battles, and a list of foughtbattles would contain just what a list of battles would contain. The ques-tion ‘Could the Battle of Waterloo have been unfought?,’ taken in oneway, is an absurd question. Yet its absurdity is something quite differentfrom the falsity that Napoleon’s strategic decisions were forced uponhim by the laws of logic.

I suspect that some of us have felt that the fatalist doctrine is unre-futed so long as no remedy has been found for the smell of logical trick-iness that hangs about such arguments as ‘Accidents can be prevented;therefore this accident could have been prevented’ or ‘I can bottle upmy laughter; therefore I could have bottled up that hoot of laughter.’For it would not have been a hoot at all, and so not that hoot, had I bot-tled up my laughter. I could not, logically, have bottled it up. For it wasan unbottled up hoot of laughter. The fact that it occurred is alreadycontained in my allusion to ‘that hoot of laughter.’ So a sort of contra-diction is produced when I try to say that that hoot of laughter need nothave occurred. No such contradiction is produced when I say ‘I did nothave to hoot with laughter.’ It is the demonstrative word ‘that . . .’which refused to consort with ‘. . . did not occur’ or ‘. . . might not haveoccurred.’

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This point seems to me to bring out an important difference betweenanterior truths and posterior truths, or between prophecies and chroni-cles. After 1815 there could be true and false statements mentioningthe Battle of Waterloo in the past tense. After 1900 there could be trueand false statements in the present and past tenses mentioning me. Butbefore 1815 and 1900 there could not be true or false statements givingindividual mention to the Battle of Waterloo or to me, and this not justbecause our names had not yet been given, nor yet just because no onehappened to be well enough equipped to predict the future in very greatdetail, but for some more abstruse reason. The prediction of an eventcan, in principle, be as specific as you please. It does not matter if infact no forecaster could know or reasonably believe his prediction tobe true. If gifted with a lively imagination, he could freely concoct astory in the future tense with all sorts of minutiae in it and this elaboratestory might happen to come true. But one thing he could not do—log-ically and not merely epistemologically could not do. He could not getthe future events themselves for the heroes or heroines of his story,since while it is still an askable question whether or not a battle will befought at Waterloo in 1815, he cannot use with their normal force thephrase ‘the Battle of Waterloo’ or the pronoun ‘it.’ While it is still anaskable question whether my parents are going to have a fourth son,he cannot use as a name the name ‘Gilbert Ryle’ or use as a pronoundesignating their fourth son the pronoun ‘he.’ Roughly, statements inthe future tense cannot convey singular, but only general propositions,where statements in the present and past tense can convey both. Morestrictly, a statement to the effect that something will exist or happen is,in so far, a general statement. When I predict the next eclipse of themoon, I have indeed got the moon to make statements about, but I havenot got her next eclipse to make statements about. Perhaps this is whynovelists never write in the future tense, but only in the past tense. They



could not get even the semblances of heroes or heroines into propheticfiction, since the future tense of their would-be-prophetic mock-narrativeswould leave it open for their heroes and heroines not to be born. But asmy phrase ‘I have not got it to make statements about’ stirs up a nest oflogical hornets, I shall bid farewell for the present to this matter.

I have chosen to start with this particular dilemma for moderatelysustained discussion for two or three connected reasons. But I did notdo so for the reason that the issue is or ever has been of paramount im-portance in the Western world. No philosopher of the first or secondrank has defended fatalism or been at great pains to attack it. Neitherreligion nor science wants it. Right-wing and Left-wing doctrines bor-row nothing from it. On the other hand we do all have our fatalist mo-ments; we do all know from inside what it is like to regard the courseof events as the continuous unrolling of a scroll written from the be-ginning of time and admitting of no additions or amendments. Yetthough we know what it is like to entertain this idea, still we are unim-passioned about it. We are not secret zealots for it or secret zealotsagainst it. We are, nearly all of the time, though also aware that the ar-gument for them is hard to rebut, cheerfully sure that the fatalist con-clusions are false. The result is that we can study the issue in the spiritof critical playgoers, not that of electors whose votes are being so-licited. It is not a burning issue. This is one reason why I have startedwith it.

Next, so little has the issue been debated by Western thinkers that Ihave been free to formulate for myself not only what seem to me thefalse steps in the fatalist argument from antecedent truth, but even thatargument itself. I have not had to recapitulate a traditional controversybetween philosophical schools, since there has been next to no suchcontroversy, as there have, notoriously, existed protracted controversiesabout Predestination and Determinism. You know, from inside your

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own skins, all that needs to be known about the issue. There are nocards of erudition up my sleeve.

Thirdly, the issue is in a way a very simple one, a very importantone and an illuminatingly tricky one. It is simple in that so few pivot-concepts are involved—just, in the first instance, the untechnical con-cepts of event, before and after, truth, necessity, cause, prevention, faultand responsibility—and of course we all know our ways about inthem—or do we? They are public highway concepts, not craftsmen’sconcepts; so none of us can get lost in them—or can we? It is importantin that if the fatalist conclusion were true, then nearly the whole of ournormal religious, moral, political, historical, scientific and pedagogicthinking would be on entirely the wrong lines. We cannot shape theworld of tomorrow, since it has already been shaped once and for all.It is a tricky issue because there is not any regulation or argumentativemanoeuvre by which it can be settled. I have produced quite an appa-ratus of somewhat elaborate arguments, all of which need expansionand reinforcement. I expect that the logical ice is pretty thin under someof them. It would not trouble me if the ice broke, since the stamp ofthe foot which broke it would itself be a partially decisive move. Buteven this move would not be the playing of any regulation logical ma-noeuvre. Such regulation manoeuvres exist only for dead philosophicalissues. It was their death which promoted the decisive moves up to thestatus of regulation manoeuvres.

Now for some general morals which can be drawn from the exis-tence of this dilemma and from attempts to resolve it. It arose out oftwo seemingly innocent and unquestionable propositions, propositionswhich are so well embedded in what I may vaguely call ‘commonknowledge’ that we should hardly wish to give them the grand title of‘theories.’ These two propositions were, first, that some statements inthe future tense are or come true, and, second, that we often can and



sometimes should secure that certain things do happen and that certainother things do not happen. Neither of these innocent-seeming propo-sitions is as yet a philosopher’s speculation, or even a scientist’s hy-pothesis or a theologian’s doctrine. They are just platitudes. We should,however, notice that it would not very often occur to anyone to statethese platitudes. People say of this particular prediction that it was ful-filled and of that particular guess that it turned out correct. To say thatsome statements in the future tense are true is a generalization of theseparticular concrete comments. But it is a generalization which there isnot usually any point in propounding. Similarly people say of particularoffences that they ought not to have been committed and of particularcatastrophes that they could or could not have been prevented. It is rel-atively rare to stand back and say in general terms that people some-times do wrong and that mishaps are sometimes our own fault. Nonethe less, there are occasions, long before philosophical or scientificspeculations begin, on which people do deliver generalities of thesesorts. It is part of the business of the teacher and the preacher, of thejudge and the doctor, of Solon and Æsop, to say general things, withconcrete examples of which everyone is entirely familiar. In one waythe generality is not and cannot be news to anyone that every day hasits yesterday and every day has its tomorrow; and yet, in another way,this can be a sort of news. There was the first occasion on which thisgenerality was presented to us, and very surprising it was—despite thefact that on every day since infancy we had thought about its particularyesterday and its particular tomorrow. There is, anyhow at the start, animportant sort of unfamiliarity about such generalizations of the totallyfamiliar. We do not yet know how we should and how we should notoperate with them, although we know quite well how to operate withthe daily particularities of which they are the generalizations. We makeno foot-faults on Monday morning with ‘will be’ and ‘was’; but when

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required to deal in the general case with the notions of the future andthe past, we are no longer sure of our feet.

The two platitudes from which the trouble arose are not in directconflict with one another. It is real or seeming deductions from the onewhich quarrel with the other, or else with real or seeming deductionsfrom it. They are not rivals such that before these deductions had beennoticed anyone would want to say ‘I accept the proposition that somestatements in the future tense are fulfilled, so naturally I reject theproposition that some things need not and should not have happened.’It is because the former proposition seems indirectly to entail that whatis was from all eternity going to be and because this, in its turn, seemsto entail that nothing is anybody’s fault, that some thinkers have feltforced to make a choice between the two platitudes. Aristotle, for ex-ample, rejected, with reservations, the platitude that statements in thefuture tense are true or false. Certain Stoics rejected the platitude thatwe are responsible for some things that happen. If we accept both plat-itudes, it is because we think that the fatalist deductions from ‘it wastrue . . .’ are fallacious or else that certain deductions drawn from ‘somethings are our fault’ are fallacious, or both.

But this raises a thorny general question. How is it that in their mostconcrete, ground-floor employment, concepts like will be, was, correct,must, make, prevent and fault behave, in the main, with exemplarydocility, but become wild when employed in what are mere first-floorgeneralizations of their ground-floor employments? We are in very littledanger of giving or taking the wrong logical change in our daily mar-keting uses of ‘tomorrow’ and ‘yesterday.’ We know perfectly well howto make our daily sales and purchases with them. Yet in the generalcase, when we try to negotiate with ‘what is,’ ‘what is to be,’ ‘whatwas’ and ‘what was to be’ we very easily get our accounts in a muddle.We are quite at home with ‘therefore’ and all at sea with ‘necessary.’



How is it that we get our accounts in a muddle when we try to do whole-sale business with ideas with which in retail trade we operate quite ef-ficiently every day of our lives? Later on I hope to give something of ananswer to this question. For the moment I merely advertise it.

Meanwhile there is another feature of the issue to which we shouldattend. I have indicated that the quandary, though relatively simple,does depend upon a smallish number of concepts, namely, in the firstinstance, upon those of event, before and after, truth, necessity, cause,prevention, fault and responsibility. Now there is not just one of theseconcepts which is the logical trouble-maker. The trouble arises out ofthe interplay between all of them. The litigation between the two initialplatitudes involves a whole web of conflicting interests. There is notjust a single recalcitrant knot in the middle of one of the concepts in-volved. All the strings between all of them are implicated in the onetangle.

I mention this point because some people have got the idea fromsome of the professions though not, I think, the practices of philoso-phers, that doing philosophy consists or should consist of untying log-ical knots one at a time—as if, to burlesque the idea, it would have beenquite proper and feasible for Hume on Monday to analyse the use ofthe term ‘cause,’ and then on Tuesday, Wednesday and Thursday tomove on to analyse seriatim the uses of the terms ‘causeway,’ ‘cautery’and ‘caution,’ in alphabetical order.

I have no special objection to or any special liking for the fashionof describing as ‘analysis’ the sort or sorts of conceptual examinationwhich constitute philosophizing. But the idea is totally false that thisexamination is a sort of garage inspection of one conceptual vehicleat a time. On the contrary, to put it dogmatically, it is always a traffic-inspector’s examination of a conceptual traffic-block, involving at least

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two streams of vehicles hailing from the theories, or points of view orplatitudes which are at cross-purposes with one another.

One other point arises in connexion with this last one. The child canbe taught a lot of words, one after another; or, when consulting the dic-tionary to find out the meanings of some unfamiliar words in a difficultpassage, he can look up these words separately in alphabetical or anyother order. This fact, among others, has encouraged the notion that theideas or concepts conveyed by these words are something like sepa-rately movable and examinable chessmen, coins, counters, snapshots—or words. But we should not think of what a word conveys as if it were,like the word, a sort of counter, though unlike the word, an invisiblecounter. Consider a wicket-keeper. He is an individual, who can befetched out of the team and separately interviewed, photographed ormassaged. But his role in the game, namely the wicket-keeping that hedoes, so interlocks with what the other cricketers do, that if theystopped playing, he could not go on keeping wicket. He alone performshis particular role, yet he cannot perform it alone. For him to keepwicket, there must be a wicket, a pitch, a ball, a bat, a bowler and abatsman. Even that is not enough. There must be a game in progressand not, for example, a funeral, a fight or a dance; and the game mustbe a game of cricket and not, for example, a game of ‘Touch Last.’ Thesame man who keeps wicket on Saturday may play tennis on Sunday.But he cannot keep wicket in a game of tennis. He can switch from oneset of sporting functions to another, but one of his functions cannot beswitched to the other game. In much the same way, concepts are notthings, as words are, but rather the functionings of words, as keepingwicket is the functioning of the wicket-keeper. Very much as the func-tioning of the wicket-keeper interlocks with the functioning of thebowler, the batsman and the rest, so the functioning of a word interlocks



with the functioning of the other members of the team for which thatword is playing. One word may have two or more functions; but oneof its functions cannot change places with another.

Let me illustrate. A game like Bridge or Poker has a fairly elaborateand well-organized technical vocabulary, as in different degrees havenearly all games, crafts, professions, hobbies and sciences. Naturallythe technical terms peculiar to Bridge have to be learned. How do welearn them? One thing is clear. We do not and could not master the useof one of them without yet having begun to learn the use of any of theothers. It would be absurd to try to teach a boy how to use the conceptof cross-ruff, without yet having introduced him to the notions of fol-lowing suit, trump and partner. But if he has been introduced to theway these terms function together in Bridge talk, then he has begun tolearn some of the elements of Bridge. Or consider the technical dictionsof English lawyers. Could a student claim to understand one or sevenof its specialist terms, though knowing nothing of the law? or claim toknow the law while not understanding at least a considerable fractionof its terminological apparatus? The terminological apparatus of a sci-ence is in the same way a team and not a mere mob of terms. The partplayed by one of them belongs, with the parts played by the others, tothe particular game or work of the whole apparatus. A person who hadmerely memorized the dictionary-paraphrases of a thousand technicalterms of physics or economics would not yet have begun to be a physi-cist or an economist. He would not yet have learned how to operatewith those terms. So he would not yet understand them. If he cannotyet think any of the thoughts of economic theory, he has not yet gotany of its special concepts.

What is true of the more or less highly technical terms of games, thelaw, the sciences, the trades and professions is true also, with importantmodifications, of the terms of everyday discourse. These stand to the

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terms of the specialists very much as civilians stand to the officers, non-commissioned officers and private soldiers of different units in theArmy. The rights, duties and privileges of soldiers are carefully pre-scribed; their uniforms, badges, stripes and buttons show their ranks,trades and units; drill, discipline and daily orders mould their move-ments. But civilians too have their codes, their habits, and their eti-quettes; their work, pay and taxes tend to be regular; their social circles,their apparel and their amusements, though not regimented, are prettystable. We know, too, how in this twentieth century of ours the distinc-tions between civilians and soldiers are notoriously blurred. Similarlythe line between un-technical and technical dictions is a blurred line,and one frequently crossed in both directions; and though untechnicalterms have not got their functions officially imposed upon them, theyhave their functions, privileges and immunities none the less. They re-semble civilians rather than soldiers, but most of them also resemblerate-payers rather than gipsies.

The functions of technical terms, that is, the concepts conveyed bythem, are more or less severely regimented. The kinds of interplay offunction for which they are built are relatively definite and circum-scribed. Yet untechnical terms, too, though they belong to no single or-ganized unit, still have their individual places in indefinitely manyoverlapping and intermingling milieus.

It can be appreciated, consequently, that the functions of terms be-come both narrower and better prescribed as they become more official.Their roles in discourse can be more strictly formulated as their com-mitments are reduced in number and in scope. Hence, the more exactlytheir duties come to being fixed by charters and commissions, the fur-ther they move from being philosophically interesting. The official con-cepts of Bridge generate few if any logical puzzles. Disputes could notbe settled or rubbers won if they were generated. Logical puzzles arise



especially over concepts that are uncommissioned, namely the civilianconcepts which, instead of having been conscripted and trained for justone definite and appointed niche in one organized unit, have grown upinto their special but unappointed places in a thousand uncharteredgroups and informal associations. This is why an issue like the fatalistissue, though starting with a quite slender stem, ramifies out so swiftlyinto seemingly remote sectors of human interests. The question whetherstatements in the future tense can be true swiftly opened out into,among a thousand others, the question whether anything is gained bylearning to swim.

Certain thinkers, properly impressed by the excellent logical disci-pline of the technical concepts of long-established and well consoli-dated sciences like pure mathematics and mechanics, have urged thatintellectual progress is impeded by the survival of the unofficial con-cepts of unspecialized thought; as if there were something damaginglyamateurish or infantile in the businesses and avocations of uncon-scripted civilians. Members of the Portland Club, the M.C.C., or theLaw Faculty of a University might, with even greater justice, contrasttheir own scrupulously pruned and even carpentered terms of art withthe undesigned dictions of everyday discourse. It is, of course, quitetrue that scientific, legal or financial thinking could not be conductedonly in colloquial idioms. But it is quite false that people could, evenin Utopia, be given their first lessons in talking and thinking in the termsof this or that technical apparatus. Fingers and feet are, for many specialpurposes, grossly inefficient instruments. But to replace the infant’s fin-gers and feet by pliers and pedals would not be a good plan—especiallyas the employment of pliers and pedals themselves depends upon theemployment of fingers and feet. Nor does the specialist when he comesto use the designed terms of his art cease to depend upon the conceptswhich he began to master in the nursery, any more than the driver,

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whose skill and interests are concentrated on the mechanically complexand delicate works of his car, cease to avail himself of the mechanicallycrude properties of the public highway. He could not use his car withoutusing the roads, though he could, as the pedestrian that he often is, usethese same roads without using his car.



11

Fatalism

Richard Taylor

A fatalist—if there is any such—thinks he cannot do anything aboutthe future. He thinks it is not up to him what is going to happen nextyear, tomorrow, or the very next moment. He thinks that even his ownbehavior is not in the least within his power, any more than the motionsof the heavenly bodies, the events of remote history, or the political de-velopments in China. It would, accordingly, be pointless for him to de-liberate about what he is going to do, for a man deliberates only aboutsuch things as he believes are within his power to do and to forego, orto affect by his doings and foregoings.

A fatalist, in short, thinks of the future in the manner in which weall think of the past. For we do all believe that it is not up to us whathappened last year, yesterday, or even a moment ago, that these thingsare not within our power, any more than are the motions of the heavens,the events of remote history or of China. And we are not, in fact, evertempted to deliberate about what we have done and left undone. At bestwe can speculate about these things, rejoice over them or repent, drawconclusions from such evidence as we have, or perhaps—if we are notfatalists about the future—extract lessons and precepts to apply hence-forth. As for what has in fact happened, we must simply take it as given;the possibilities for action, if there are any, do not lie there. We may,

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indeed, say that some of those past things were once within our power,while they were still future—but this expresses our attitude toward thefuture, not the past.

There are various ways in which a man might get to thinking in thisfatalistic way about the future, but they would be most likely to resultfrom ideas derived from theology or physics. Thus, if God is really all-knowing and all-powerful, then, one might suppose, perhaps he has al-ready arranged for everything to happen just as it is going to happen,and there is nothing left for you or me to do about it. Or, without bring-ing God into the picture, one might suppose that everything happensin accordance with invariable laws, that whatever happens in the worldat any future time is the only thing that can then happen, given that cer-tain other things were happening just before, and that these, in turn, arethe only things that can happen at that time, given the total state of theworld just before then, and so on, so that again, there is nothing left forus to do about it. True, what we do in the meantime will be a factor indetermining how some things finally turn out—but these things that weare going to do will perhaps be only the causal consequences of whatwill be going on just before we do them, and so on back to a not distantpoint at which it seems obvious that we have nothing to do with whathappens then. Many philosophers, particularly in the seventeenth andeighteenth centuries, have found this line of thought quite compelling.

I want to show that certain presuppositions made almost universallyin contemporary philosophy yield a proof that fatalism is true, withoutany recourse to theology or physics. If, to be sure, it is assumed thatthere is an omniscient god, then that assumption can be worked intothe argument so as to convey the reasoning more easily to the unphilo-sophical imagination, but this assumption would add nothing to theforce of the argument, and will therefore be omitted here. And similarly,certain views about natural laws could be appended to the argument,

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perhaps for similar purposes, but they, too, would add nothing to itsvalidity, and will therefore be ignored.

Presuppositions. The only presuppositions we shall need are the sixfollowing.

First, we presuppose that any proposition whatever is either true or,if not true, then false. This is simply the standard interpretation, tertiumnon datur, of the law of excluded middle, usually symbolized (p v -p),which is generally admitted to be a necessary truth.

Second, we presuppose that, if any state of affairs is sufficient for,though logically unrelated to, the occurence of some further conditionat the same or any other time, then the former cannot occur without thelatter occuring also. This is simply the standard manner in which theconcept of sufficiency is explicated. Another and perhaps better way ofsaying the same thing is that, if one state of affairs ensures without log-ically entailing the occurrence of another, then the former cannot occurwithout the latter occuring. Ingestion of cyanide, for instance, ensuresdeath under certain familar circumstances, though the two states of af-fairs are not logically related.

Third, we presuppose that, if the occurrence of any condition is nec-essary for, but logically unrelated to, the occurrence of some other con-dition at the same or any other time, then the latter cannot occur withoutthe former occurring also. This is simply the standard manner in whichthe concept of a necessary condition is explicated. Another and perhapsbetter way of saying the same thing is that, if one state of affairs is es-sential for another, then the latter cannot occur without it. Oxygen, forinstance, is essential to (though it does not by itself ensure) the main-tenance of human life—though it is not logically impossible that weshould live without it.

Fourth, we presuppose that, if one condition or set of conditions issufficient for (ensures) another, then that other is necessary (essential)

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for it, and conversely, if one condition or set of conditions is necessary(essential) for another, then that other is sufficient for (ensures) it. Thisis but a logical consequence of the second and third presuppositions.

Fifth, we presuppose that no agent can perform any given act if thereis lacking, at the same or any other time, some condition necessary forthe occurrence of that act. This follows, simply from the idea of any-thing being essential for the accomplishment of something else. I can-not, for example, live without oxygen, or swim five miles without everhaving been in water, or read a given page of print without havinglearned Russian, or win a certain election without having been nomi-nated, and so on.

And sixth, we presuppose that time is not by itself “efficacious”; thatis, that the mere passage of time does not augment or diminish the ca-pacities of anything and, in particular, that it does not enhance or de-crease an agent’s powers or abilities. This means that if any substanceor agent gains or loses powers or abilities over the course of time—such as, for instance, the power of a substance to corrode, or a man todo thirty push-ups, and so on—then such gain or loss is always the re-sult of something other than the mere passage of time.

With these presuppositions before us, we now consider two situa-tions in turn, the relations involved in each of them being identical ex-cept for certain temporal ones.

The first situation. We imagine that I am about to open my morningnewspaper to glance over the headlines. We assume, further, that con-ditions are such that only if there was a naval battle yesterday does thenewspaper carry a certain kind (shape) of headline—i.e., that such abattle is essential for this kind of headline—whereas if it carries a cer-tain different sort (shape) of headline, this will ensure that there wasno such battle. Now, then, I am about to perform one or the other oftwo acts, namely, one of seeing a headline of the first kind, or one of

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seeing a headline of the second kind. Call these alternative acts S andS' respectively. And call the propositions, “A naval battle occurred yes-terday” and “No naval battle occurred yesterday”, P and P' respectively.We can assert, then, that if I perform act S', then my doing such willensure that there was a naval battle yesterday (i.e., that P is true),whereas if I perform S', then my doing that will ensure that no suchbattle occurred (or, that P' is true).

With reference to this situation, then, let us now ask whether it is upto me which sort of headline I shall read as I open the newspaper; thatis, let us see whether the following proposition is true:

(A) It is within my power to do S, and it is also within my power todo S'.

It seems quite obvious that this is not true. For if both these acts wereequally within my power, that is, if it were up to me which one to do,then it would also be up to me whether or not a naval battle has takenplace, giving me a power over the past which I plainly do not possess.It will be well, however, to express this point in the form of a proof, asfollows:

1. If P is true, then it is not within my power to do S' (for in case Pis true, then there is, or was, lacking a condition essential for mydoing S', the condition, namely, of there being no naval battleyesterday).

2. But if P' is true, then it is not within my power to do S (for asimilar reason).

3. But either P is true, or P' is true.∴4. Either it is not within my power to do S, or it is not within my

power to do S';

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and (A) is accordingly false. A common-sense way of expressing thisis to say that what sort of headline I see depends, among other things,on whether a naval battle took place yesterday, and that, in turn, is notup to me.

Now this conclusion is perfectly in accordance with common sense,for we all are, as noted, fatalists with respect to the past. No one con-siders past events as being within his power to control; we simply haveto take them as they have happened and make the best of them. It issignificant to note, however, that, in the hypothetical sense in whichstatements of human power or ability are usually formulated, one doeshave power over the past. For we can surely assert that, if I do S, thiswill ensure that a naval battle occurred yesterday, whereas if alterna-tively, I do S', this will equally ensure the nonoccurrence of such a bat-tle, since these acts are, in terms of our example, quite sufficient forthe truth of P and P' respectively. Or we can equally say that I can en-sure the occurrence of such a battle yesterday simply by doing S andthat I can ensure its nonoccurrence simply by doing S'. Indeed, if Ishould ask how I can go about ensuring that no naval battle occurredyesterday, perfectly straightforward instructions can be given, namely,the instruction to do S' and by all means to avoid doing S. But of coursethe hitch is that I cannot do S' unless P' is true, the occurrence of thebattle in question rendering me quite powerless to do it.

The second situation. Let us now imagine that I am a naval com-mander, about to issue my order of the day to the fleet. We assume, fur-ther, that, within the totality of other conditions prevailing, my issuingof a certain kind of order will ensure that a naval battle will occur to-morrow, whereas if I issue another kind of order, this will ensure thatno naval battle occurs. Now, then, I am about to perform one or theother of these two acts, namely, one of issuing an order of the first sortor one of the second sort. Call these alternative acts O and O' respec-

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tively. And call the two propositions, “A naval battle will occur tomor-row” and “No naval battle will occur tomorrow,” Q and Q' respectively.We can assert, then, that, if I do act O, then my doing such will ensurethat there will be a naval battle, whereas if I do O', my doing that willensure that no naval battle will occur.

With reference to this situation, then, let us now ask whether it is upto me which sort of order I issue; that is, let us see whether the follow-ing proposition is true:

(B) It is within my power to do O, and it is also within my power todo O'.

Anyone, except a fatalist, would be inclined to say that, in the situ-ation we have envisaged, this proposition might well be true, that is,that both acts are quite within my power (granting that I cannot do bothat once). For in the circumstances we assume to prevail, it is, one wouldthink, up to me as the commander whether the naval battle occurs ornot; it depends only on what kind of order I issue, given all the otherconditions as they are, and what kind of order is issued is somethingquite within my power. It is precisely the denial that such propositionsare ever true that would render one a fatalist.

But we have, unfortunately, the same formal argument to show that(B) is false that we had for proving the falsity of (A), namely:

1'. If Q is true, then it is not within my power to do O' (for in case Qis true, then there is, or will be, lacking a condition essential formy doing O', the condition, namely, of there being no naval battletomorrow).

2'. But if Q' is true, then it is not within my power to do O (for asimilar reason).

11 Fatalism 129


3'. But either Q is true, or Q' is true.∴4'. Either it is not within my power to do O, or it is not within my

power to do O';

and (B) is accordingly false. Another way of expressing this is to saythat what sort of order I issue depends, among other things, on whethera naval battle takes place tomorrow—for in this situation a naval battletomorrow is (by our fourth presupposition) a necessary condition ofmy doing O, whereas no naval battle tomorrow is equally essential formy doing O'.

Considerations of time. Here it might be tempting, at first, to saythat time makes a difference, and that no condition can be necessaryfor any other before that condition exists. But this escape is closed byboth our fifth and sixth presuppositions. Surely if some condition, atany given time, whether past, present, or future, is necessary for theoccurrence of something else, and that condition does not in fact existat the time it is needed, then nothing we do can be of any avail in bring-ing about that occurrence for which it is necessary. To deny this wouldbe equivalent to saying that I can do something now which is, togetherwith other conditions prevailing, sufficient for, or which ensures, theoccurrence of something else in the future, without getting that futureoccurrence as a result. This is absurd in itself and contrary to our secondpresupposition. And if one should suggest, in spite of all this, that astate of affairs that exists not yet cannot, just because of this temporalremoval, be a necessary condition of anything existing prior to it, thiswould be logically equivalent to saying that no present state of affairscan ensure another subsequent to it. We could with equal justice saythat a state of affairs, such as yesterday’s naval battle, which exists nolonger, cannot be a necessary condition of anything existing subse-quently, there being the same temporal interval here; and this would be

130 RICHARD TAYLOR


arbitrary and false. All that is needed, to restrict the powers that I imag-ine myself to have to do this or that, is that some condition essential tomy doing it does not, did not, or will not occur.

Nor can we wriggle out of fatalism by representing this sort of situ-ation as one in which there is a simple loss of ability or power resultingfrom the passage of time. For according to our sixth presupposition, themere passage of time does not enhance or diminish the powers or abil-ities of anything. We cannot, therefore, say that I have the power to doO' until, say, tomorrow’s naval battle occurs, or the power to do O untiltomorrow arrives and we find no naval battle occurring, and so on. Whatrestricts the range of my power to do this thing or that is not the meretemporal relations between my acts and certain other states of affairs,but the very existence of those states of affairs themselves; and accord-ing to our first presupposition, the fact of tomorrow’s containing, or lack-ing, a naval battle, as the case may be, is no less a fact than yesterday’scontaining or lacking one. If, at any time, I lack the power to perform acertain act, then it can only be the result of something, other than thepassage of time, that has happened, is happening, or will happen. Thefact that there is going to be a naval battle tomorrow is quite enough torender me unable to do O', just as the fact that there has been a navalbattle yesterday renders me unable to do S', the nonoccurrence of thoseconditions being essential, respectively, for my doing those things.

Causation. Again, it does no good here to appeal to any particularanalyses of causation, or to the fact, if it is one, that causes only “work”forwards and not backwards, for our problem has been formulated with-out any reference to causation. It may be, for all we know, that causalrelations have an unalterable direction (which is an unclear claim in it-self ), but it is very certain that the relations of necessity and sufficiencybetween events or states of affairs have not, and it is in terms of thesethat our data have been described.

11 Fatalism 131


The law of excluded middle. There is, of course, one other way toavoid fatalism, and that is to deny one of the premises used to refute(B). The first two, hypothetical, premises cannot be denied, however,without our having to reject all but the first, and perhaps the last, of ouroriginal six presuppositions, and none of these seems the least doubtful.And the third premise—that either Q is true, or Q' is true—can be de-nied only by rejecting the first of our six presuppositions, that is, by re-jecting the standard interpretation, tertium non datur, of what is calledthe law of excluded middle.

This last escape has, however, been attempted, and it apparently in-volves no absurdity. Aristotle, according to an interpretation that issometimes rendered of his De Interpretation, rejected it. According tothis view, the disjunction (Q v Q') or, equivalently, (Q v −Q), which isan instance of the law in question, is a necessary truth. Neither of itsdisjuncts, however—i.e., neither Q, nor Q'—is a necessary truth nor,indeed, even a truth, but is instead a mere “possibility,” or “contin-gency” (whatever that may mean). And there is, it would seem, no ob-vious absurdity in supposing that two propositions, neither of them trueand neither of them false, but each “possible,” might nevertheless com-bine into a disjunction which is a necessary truth—for that disjunctionmight, as this one plainly does, exhaust the possibilities.

Indeed, by assuming the truth of (B)—i.e., the statement that it iswithin my power to do O and it is also within my power to do O'—andsubstituting this as our third premise, a formal argument can be ren-dered to prove that a disjunction of contradictories might disjoin propo-sitions which are neither true nor false. Thus:

1". If Q is true, then it is not within my power to do O'.2". But if Q' is true, then it is not within my power to do O.

132 RICHARD TAYLOR


3". But it is within my power to do O, and it is also within my powerto do O'.

∴4". Q' is not true, and Q is not true;

and to this we can add that, since Q and Q' are logical contradictories,such that if either is false then the other is true, then Q is not false, andQ' is not false—i.e., that neither of them is true and neither of themfalse.

There seems to be no good argument against this line of thoughtwhich does not presuppose the very thing at issue, that is, which doesnot presuppose, not just the truth of a disjunction of contradictories,which is here preserved, but one special interpretation of the law thusexpressed, namely, that no third value, like “possible,” can ever be as-signed to any proposition. And that particular interpretation can, per-haps, be regarded as a more or less arbitrary restriction.

We would not, furthermore, be obliged by this line of thought to re-ject the traditional interpretation of the so-called law of contradiction,which can be expressed by saying that, concerning any proposition, notboth it and its contradictory can be true—which is clearly consistentwith what is here suggested.

Nor need we suppose that, from a sense of neatness and consistency,we ought to apply the same considerations to our first situation and toproposition (A)—that, if we so interpret the law in question as to avoidfatalism with respect to the future, then we ought to retain the same in-terpretation as it applies to things past. The difference here is that wehave not the slightest inclination to suppose that it is at all within ourpower what happened in the past, or that propositions like (A) in situ-ations such as we have described are ever true, whereas we do, if weare not fatalists, believe that it is sometimes within our power what

11 Fatalism 133


happens in the future, that is, that propositions like (B) are sometimestrue. And it was only from the desire to perserve the truth of (B), butnot (A), and thus avoid fatalism, that the tertium non datur was doubted,using (B) as a premise.

Temporal efficacy. It now becomes apparent, however, that if weseek to avoid fatalism by this device, then we shall have to reject notonly our first but also our sixth presupposition; for on this view timewill by itself have the power to render true or false certain propositionswhich were hitherto neither, and this is an “efficacy” of sorts. In fact,it is doubtful whether one can in any way avoid fatalism with respectto the future while conceding that things past are, by virtue of theirpastness alone, no longer within our power without also conceding anefficacy to time; for any such view will entail that future possibilities,at one time within our power to realize or not, cease to be such merelyas a result of the passage of time—which is precisely what our sixthpresupposition denies. Indeed, this is probably the whole point in cast-ing doubt upon the law of excluded middle in the first place, namely,to call attention to the status of some future things as mere possibilities,thus denying both their complete factuality and their complete lack ofit. If so, then our first and sixth presuppositions are inseparably linked,standing or falling together.

The assertion of fatalism. Of course one other possibility remains,and that is to assert, out of a respect for the law of excluded middle anda preference for viewing things under the aspect of eternity, that fatal-ism is indeed a true doctrine, that propositions such as (B) are, like (A),never true in such situations as we have described, and that the differ-ence in our attitudes toward things future and past, which leads us tocall some of the former but none of the latter “possibilities,” results en-tirely from epistemological and psychological considerations—such

134 RICHARD TAYLOR


as, that we happen to know more about what the past contains thanabout what is contained in the future, that our memory extends to pastexperiences rather than future ones, and so on. Apart from subjectivefeelings of our power to control things, there seem to be no good philo-sophical reasons against this opinion, and very strong ones in its favor.

11 Fatalism 135


12

Time, Truth, and Ability

Richard Taylor and Steven M. Cahn

We shall here be concerned with statements of the form ‘M does A at t,’wherein M designates a specific person, A a specific action and t a spe-cific time. We shall refer to these as R-statements. Thus, ‘Someoneraised his hand at noon last Tuesday,’ ‘Stilpo raised his hand,’ and ‘Stilpodid something at noon last Tuesday’ are none of them R-statements; but‘Stilpo raised his right hand at noon last Tuesday’ is an R-statement.

Let us assume that it sometimes at least makes sense to speak of anagent’s being able to render an R-statement true, as distinguished, forexample, from simply discovering that it is true; and similarly, that itsometimes makes sense to speak of his being able to render an R-state-ment false. Thus, Stilpo could render it true that he is running at a cer-tain time simply by running at that time, and this would be somethingquite different from his then merely discovering—observing, noting,etc.—that he is running. He could, of course, render the same statementfalse in a variety of ways—by standing still, for instance, or by lyingdown, and so on. We, on the other hand, could not in any similar wayrender that R-statement true. We could only discover by some meansthat it is true, or that it is false—by looking at Stilpo at the time in ques-tion, for instance, to see whether he is then running.

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Further, let us assume that it sometimes at least makes sense to speakof asking someone to render an R-statement true. This, of course, isonly an application of the general principle that, in the case of some-thing that someone is able to do, it sometimes makes sense to ask himto do it. To illustrate, suppose that Crates has a bet with Metrocles thatStilpo will pass through the Diomean Gate at noon on the followingday (call that day D). Now it surely seems to make sense that Cratesmight ask, and perhaps even bribe, Stilpo to do just that—to pick justthat time to pass through the gate—and thus render true the R-statement‘Stilpo passes through the Diomean Gate at noon on day D.’ That a re-quest or even a bribe would not be out of place in such circumstancessuggests both that it sometimes makes sense to speak of an agent’s ren-dering an R-statement true, and that it sometimes makes sense to asksomeone—namely, the agent referred to in such a statement—to do it.

Now it is easy enough to state, in general terms, what one has to doin order to render a given R-statement true. He has to do precisely whatthe statement in question says he does, at precisely the time the state-ment says he does it. The only way Stilpo can render it true that hepasses through the Diomean Gate at a specific time is to pass throughthe gate at just that time. Similarly, the only way he can render it falseis to refrain from passing through the gate at just that time. For someoneto be able to render an R-statement true, then, consists simply of hisbeing able to do something which is logically both necessary and suf-ficient for the truth of the statement to the effect that he does the thingin question at the time in question. Nothing else suffices, and this willneed to be borne in mind.

Finally, we shall assume that, in case one speaks truly in uttering aparticular R-statement at a particular time, then one also speaks trulyin uttering the same R-statement at any other time. If, for example,one were to speak truly in saying that Stilpo is running at noon on a

138 RICHARD TAYLOR AND STEVEN M. CAHN


given Tuesday, one would also speak truly if one said the same thingagain a week later, or at any other time. This, of course, is only an ap-plication of the orthodox assumption that complete statements, or theutterances of them, are not converted from true to false, or from falseto true, just by the passage of time. There are some statements, to besure, like ‘Stilpo is running,’ which are not, as they stand, true everytime they are uttered, since it is not always the case that Stilpo is run-ning. But that is not an R-statement. If one adds to it an explicit refer-ence to the time at which Stilpo is alleged to be running—say, at noonon a given Tuesday—then it becomes an R-statement. It also therebybecomes a statement that is true every time it is uttered, in case it istrue at all, for one can on Wednesday still say truly that Stilpo wasrunning at noon on the day preceding, in case he was, even thoughStilpo may in the meantime have stopped running. This assumption,it should be noted, does not imply that truth and falsity are ‘properties’of ‘propositions’ that might be gained or lost through the passage oftime, nor does it imply that they are not. Some say that they are, othersthat they are not, and still others that such a notion is meaningless tobegin with; but we, at least, prefer to take no stand on that somewhatmetaphysical point.

Now let us consider three times, t1, t2 and t3, all of them being past,and t1 being earlier than t2 which is earlier than t3. Consider, then, theR-statement (S):

Stilpo walks through the Diomean Gate at t2

and assume that statement, tenselessly expressed so as to avoid am-biguity in what follows, to be true. What we want to consider is: which,if any, of the following, which are not R-statements but are statementsconcerning Stilpo’s abilities, are also true?

12 Time, Truth, and Ability 139


1. Stilpo was at t3 able to render S false.2. Stilpo was at t3 able to render S true.3. Stilpo was at t1 able to render S false.4. Stilpo was at t1 able to render S true.5. Stilpo was at t2 able to render S false.6. Stilpo was at t2 able to render S true.

Now the first of these is quite evidently false. If, as assumed, it istrue that Stilpo was walking through the gate at t2, then there is ab-solutely nothing he (or anyone) was able to do at t3 which could renderthat statement false. It was, it would seem natural to say, by that timetoo late for that. He was perhaps able at t3 to refrain from passingthrough the gate again, of course, and he was perhaps able to regretthat he had walked through it, to wish he had not, and so on, but hisdoing any of those things would not have the least tendency to renderS false. Or we might think that he was at t3 able to find conclusive ev-idence that he had not walked through the gate; but that is not in factanything that he was able to do, for he had already walked through thegate, and hence there was at t3 no conclusive evidence to the contrarythat he could possibly find.

The second statement seems also to be clearly false. S is, we said,true. So if anyone were, at t2 (or any other time), to assert S, he wouldthen be speaking truly. No sense, then, can be made of Stilpo’s subse-quently undertaking to render it true. It is in this case not only too latefor him to do anything about that; it is also superfluous. What he wantsto do—to render S true—he has already done.

The truth or falsity of the third statement is not quite so obvious, butit certainly appears to be false, and for the same kind of reason that (1)is false. That is, if it is true that Stilpo was walking through the gate att2, then it is difficult to see what he (or anyone) was able to do at t1



which might render that statement false. (Analogously to the foregoingremarks one might say, though it seems less natural to do so, that it wasat that time “too early” for that. Stilpo was perhaps able at t1 to refrainfrom then and there walking through the gate, to be sure, and perhapshe did then refrain, but that does not in the least affect the truth of S,which says nothing about what he was doing at t1. Or we might thinkthat he was, at t1, able to find some conclusive evidence or indicationthat he was not going to walk through the gate at t2, but that again isnot anything he was able to do; for he did walk through the gate at t2,and hence there was at t1 no conclusive indication to the contrary thathe could possibly have found.

To have been able at t1 to render S false, Stilpo would have to havebeen able at t1 to do something that would have been logically sufficientfor the falsity of S. But nothing that he might have done at t1 has theleast logical relevance to the truth or falsity of S. We might, to be sure,suppose that he was able at t1 firmly to resolve not to walk through thegate at t2, but his making such a resolve would not be sufficient for thefalsity of S. In fact, it has no logical relevance to S, which is, in anycase, true.

Perhaps, then, Stilpo was able at t1 to do something which wouldhave been causally or physically sufficient for the falsity of S—to com-mit suicide, for example. Actually, this suggestion is irrelevant, for wehave said that one renders an R-statement false only by doing some-thing that is logically sufficient for its falsity. But even if it were rele-vant, it would not do. What is behind this suggestion is, obviously, thatit is physically impossible that Stilpo should be walking through thegate at t2 in case he killed himself at t1. This is of course true—but ifso, then it is also true that it was physically impossible that Stilposhould have killed himself at t1 in case he was walking through the gateat t2—and we have said from the start that he was then walking through



the gate. The only conclusion, then, is that (3) is false, even on this en-larged and still irrelevant conception of what is involved in renderingan R-statement false.

The fourth statement appears false for reasons similar to those givenfor the falsity of (2). Namely, that S is true, or such that if anyone haduttered S at t1 he would then have spoken truly. No sense, then, can bemade of Stilpo’s being able to do something at t1 to render it true. Therewould have been no point, for example, in his passing through the gateat t1, for that would certainly not by itself render it true that he was stillpassing through the gate at t2. Similarly, it would not have been enoughfor him simply to have resolved at t1 to pass through the gate at t2, forthat would have been entirely compatible with the falsity of S, whichin any case neither says nor implies anything whatsoever about Stilpo’sresolutions. People do not always act upon their resolves anyway, andthere is in any case no logical necessity in their doing so. Besides, any-thing Stilpo might do at t1 would be superfluous, even if it were notpointless, for one can no more render true a statement that is true thanhe can render hard a piece of clay that is hard. He can only verify thatit is true, and this, we have seen, is something quite different. AnythingStilpo does at t1, or is able to do then, is entirely wasted.

The fifth statement likewise appears to be false. If, as we are assum-ing, it is true that Stilpo was passing through the gate at t2, then it isquite impossible to see what he might be able then and there to do, inaddition to passing through the gate, which would, if done, render thatstatement false. Indeed, it is logically impossible that there should beany such supplementary action, for no matter what it was, it would haveno tendency to render S false. Even if Stilpo were to declare, mostgravely and emphatically, that he was not passing through the gate, thiswould not render it false that he was—it would only render him a liar.A condition logically sufficient for the truth of S—namely, Stilpo’swalking through the gate—already obtains at t2, and can by no means



be conjoined with another condition logically sufficient for the falsityof that statement. Now Stilpo might, to be sure, suddenly stop walkingthrough the gate, which we can for now assume that he is able to do,but this would not in the least alter the truth of S. On the contrary, unlesshe were walking through the gate at t2, and unless, accordingly, S weretrue, he could not then stop walking. His ceasing to walk would onlyrender it false that he was walking shortly after t2; and this is hardlyinconsistent with S.

The sixth statement, finally, appears, unlike the others, to be quiteevidently true in one seemingly trivial sense, but nonsensical in another.The sense in which it is true is simply this: that if S is true, then it fol-lows that Stilpo was able to be walking through the gate at t2, thatbeing, in fact, precisely what he was doing. It is not clear, however,what sense can be attached to his being able to render true what is true,just as it is not clear what sense could be made of someone’s renderinghard some clay that is already hard.

If a piece of clay is hard we cannot sensibly ask anyone to render ithard. Similarly, if Stilpo is walking through the gate we cannot sensiblyask him to render it true that he is walking through the gate. We cannotsensibly ask him to be walking through the gate, for he is already doingthat, and our request would be otiose and absurd, like asking a manwho is sitting to be sitting, or one who is talking to be talking. We can-not ask him to continue walking through the gate, for that would notbe to the point. It would, if done, only render it true that he was stillwalking through the gate at some time after t2, which is not what weare after. And obviously, there is nothing else we could ask him to dowhich is anywhere to the point.

The only conclusion we can draw is that, of the six statements beforeus, those that make clear sense are all false, and the only one that istrue makes only trivial and dubious sense. More generally, we can saythat while it might, as we assumed at the beginning, make sense to



speak of being able to render an R-statement true, or being able to ren-der such a statement false, people can in fact only render true those R-statements that are true, and can only render false those that are false,and that these latter two conceptions themselves make very dubioussense.1

NOTES

1. I believe this paradoxical conclusion follows from the mistaken as-sumption that every true state has always been true. For a full-scale attackon this supposition, see my Fate, Logic, and Time, originally published in1967 by Yale University Press and reprinted in 2004 by Wipf and Stock Pub-lishers [Steven M. Cahn].



QUESTIONS

IV. LOGIC AND FREEDOM

1. If, as Ryle supposes, I guess that a horse named Eclipse will win a race,and then Eclipse does win the race, was my guess true at the time I made it?

2. Do you doubt any of Taylor’s six presuppositions, do you question hisreasoning, or do you accept his conclusion?

3. Do you agree with Taylor and Cahn that you cannot render true a state-ment that is false?

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V

LOGIC AND REALITY

How are the rules of deduction to be justified? Any argumentfor them presumes them, and so begs the question. Susan Haackidentifies two responses to this quandary: the pessimistic ap-proach that views deduction as unjustified and the optimistic ap-proach that views deduction as needing no justification.

How are counterfactual claims to be understood? Much ofour everyday reasoning involves judgments of the form, had poccurred, then q would have resulted. Nelson Goodman surveysand tries to resolve the many problems that such conditionalspose.

How is the existential quantifier to be understood? For ex-ample, does the statement Pegasus does not exist seem to imply,paradoxically, that there is a Pegasus? Willard V. Quine rejectsthis view, arguing that “To be is to be the value of a variable.”


13

The Justification of Deduction

Susan Haack

(1) It is often taken for granted by writers who propose—and, for thatmatter, by writers who oppose—‘justifications’ of induction, that de-duction either does not need, or can readily be provided with, justifi-cation. The purpose of this paper is to argue that, contrary to thiscommon opinion, problems analogous to those which, notoriously,arise in the attempt to justify induction, also arise in the attempt to jus-tify deduction.

Hume presented us with a dilemma: we cannot justify induction de-ductively, because to do so would be to show that whenever the pre-misses of an inductive argument are true, the conclusion must be truetoo—which would be too strong; and we cannot justify induction in-ductively, either, because such a ‘justification’ would be circular. I pro-pose another dilemma: we cannot justify deduction inductively, becauseto do so would be, at best, to show that usually, when the premisses ofa deductive argument are true, the conclusion is true too—which wouldbe too weak; and we cannot justify deduction deductively, either, be-cause such a justification would be circular.

The parallel between the old and the new dilemmas can be illustratedthus:

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(2) A necessary preliminary to serious discussion of the problems ofjustifying induction/deduction is a clear statement of them.

This means, first, giving some kind of characterisation of ‘inductiveargument’ and ‘deductive argument.’ This is a more difficult task thanseems to be generally appreciated. It will hardly do, for example, tocharacterise deductive arguments as ‘non-ampliative’ (Salmon [1966])or ‘explicative’ (Barker [1965]), and inductive arguments as ‘ampliative’or ‘non-explicative’; for these characteristics are apt to turn out eitherfalse, if the key notion of ‘containing nothing in the conclusion not al-ready contained in the premisses’ is taken literally, or trivial, if it is not.

Because of the difficulties of demarcating ‘inductive’ and ‘deduc-tive’ inference, it seems more profitable to define an argument:

An argument is a sequence A1 . . . An of sentences (n ≥ I), ofwhich A1 . . . An – 1 are the premisses and An is the conclusion—and

150 SUSAN HAACK

Hume’s dilemmainduction

The new dilemmadeduction

deductivejustification—too strong

inductivejustification—circular

inductivejustification—too weak

deductivejustification—circular


then to try to distinguish inductive from deductive standards of a‘good argument.’

It is well known that deductive standards of validity may be put ineither of two ways: syntactically or semantically. So:

D1 An argument A1 . . . An–1 ├ An is deductively valid (in LD) just incase the conclusion, An, is deducible from the premisses, A1 . . .An–1, and the axioms of LD, if any, in virtue of the rules ofinference of LD (the syntactic definition).

D2 An argument A1 . . . An–1 ├ An is deductively valid just in case itis impossible that the premisses, A1 . . . An–1, should be true, andthe conclusion, An, false (the semantic definition).

Similarly, we can express standards of inductive strength either syn-tactically or semantically; the syntactic definition would follow D1 butwith ‘L1’ for ‘LD’; the semantic definition would follow D2 but with‘it is improbable, given that the premisses are true, that the conclusionis false.’

The question now arises, which of these kinds of characterisationshould we adopt in our statement of the problems of justifying deduc-tion/induction? This presents a difficulty. If we adopt semantic accountsof deductive validity/inductive strength, the problem of justificationwill seem to have been trivialised. The justification problem will reap-pear, however, in a disguised form, as the question ‘Are there any de-ductively valid/inductively strong arguments?’. If, on the other hand,we adopt syntactic accounts of deductive validity/inductive strength,the nature of the justification problem is clear: to show that argumentswhich are deductively valid/inductively strong are also truth-preserving/truth-preserving most of the time (i.e. deductively valid/inductivelystrong on the semantic accounts). On the other hand, there is the diffi-culty that we must somehow specify which systems are possible values

13 The Justification of Deduction 151


of ‘LD’ and ‘L1,’ and this will presumably require appeal to inevitablyvague considerations concerning the intentions of the authors of a for-mal system.

A convenient compromise is this. There are certain forms of infer-ence, such as the rule:

R1 From: m/n of all observed F’s have been G’sto infer: m/n of all F’s are G’s

which are commonly taken to be inductively strong, and similarly, cer-tain forms of inference, such as

MPP From: A ⊃ B, Ato infer: B

which are generally taken to be deductively valid. Analogues of thegeneral justification problems can now be set up as follows:

the problem of the justification of induction: show that RI istruth-preserving most of the time.the problem of the justification of deduction: show that MPP istruth-preserving.

My procedure will be, then, to show that difficulties arise in the attemptto justify MPP which are analogous to notorious difficulties arising inthe attempt to justify RI.

(3) I consider first the suggestion that deduction needs no justifica-tion, that the call for a proof that MPP is truth-preserving is somehowmisguided.

152 SUSAN HAACK


An argument for this position might go as follows:

It is analytic that a deductively valid argument is truth-preserving,for by ‘valid’ we mean ‘argument whose premisses could not betrue without its conclusion being true too.’ So there can be noserious question whether a deductively valid argument is truth-preserving.

It seems clear enough that anyone who argued like this would be thevictim of a confusion. Agreed, if we adopt a semantic definition of ‘de-ductively valid’ it follows immediately that deductively valid argumentsare truth-preserving. But the problem was, to show that a particular formof argument, a form deductively valid in the syntactic sense, is truth-preserving; and this is a genuine problem, which has simply beenevaded. Similar arguments show the claim, made e.g. by Strawson in[1952], p. 257, that induction needs no justification, to be confused.

(4) I argued in Section (1) that ‘justifications’ of deduction are liableeither to be inductive and too weak, or to be deductive and circular.The former, inductive kind of justification has enjoyed little popularity(except with the Intuitionists? cf. Brouwer [1952]). But arguments ofthe second kind are not hard to find.

(a) Consider the following attempt to justify MPP:

A1 Suppose that ‘A’ is true, and that ‘A ⊃ B’ is true. By the truth-table for ‘⊃,’ if ‘A’ is true and ‘A ⊃ B’ is true, then ‘B’ is true too.So ‘B’ must be true too.

This argument has a serious drawback: it is of the very form whichit is supposed to justify. For it goes:



A1' Suppose C (that ‘A’ is true and that ‘A ⊃ B’ is true). If C then D(if ‘A’ is true and ‘A ⊃ B’ is true, ‘B’ is true). So, D (‘B’ is truetoo).

The analogy with Black’s ‘self-supporting’ argument for induction[1954] is striking. Black proposes to support induction by means of theargument:

A2 RI has usually been successful in observed instances.∴ RI is usually successful.

He defends himself against the charge of circularity by pointing outthat this argument is not a simple case of question-begging: it does notcontain its conclusion as a premiss. It might, similarly, be pointed outthat A1' is not a simple case of question-begging: for it does not containits conclusion as a premiss, either.

One is inclined to feel that A2 is objectionably circular, in spite ofBlack’s defence; and this intuition can be supported by an argument,like Salmon’s [1966], to show that if A2 supports RI, an exactly analo-gous argument would support a counter-inductive rule, say:

RCI From: most observed F’s have not been G’sto infer: most F’s are G’s.

Thus,

A3 RCI has usually been unsuccessful in the past.∴ RCI is usually successful.

In a similar way, one can support the intuition that there is somethingwrong with A1', in spite of its not being straightforwardly question-

154 SUSAN HAACK


begging, by showing that if A1' supports MPP, an exactly analogous ar-gument would support a deductively invalid rule, say:

MM (modus morons);From: A ⊃ B and Bto infer: A.

Thus:

A4 Supposing that ‘A ⊃ B’ is true and ‘B’ is true, ‘A ⊃ B’ is true⊃ ‘B’ is true.Now, by the truth-table for ‘⊃,’ if ‘A’ is true, then, if ‘A ⊃ B’ istrue, ‘B’ is true. Therefore, ‘A’ is true.

This argument, like A1, has the very form which it is supposed to justify.For it goes:

A4' Suppose D (if ‘A ⊃ B’ is true, ‘B’ is true).If C, then D (if ‘A’ is true, then, if ‘A ⊃ B’ is true, ‘B’ is true).So, C (‘A’ is true).

It is no good to protest that A4' does not justify modus morons becauseit uses an invalid rule of inference, whereas A1' does justify modus po-nens, because it uses a valid rule of inference—for to justify our con-viction that MPP is valid and MM is not is precisely what is at issue.

Neither is it any use to protest that A1' is not circular because it is anargument in the meta-language, whereas the rule which it is supposedto justify is a rule in the object language. For the attempt to save theargument for RI by taking it as a proof, on level 2, of a rule of level 1,also falls prey to the difficulty that we could with equal justice give acounter-inductive argument, on level 2, for the counter-inductive rule



at level 1. And similarly, if we may give an argument using MPP, atlevel 2, to support the rule MPP at level 1, we could, equally, give anargument, using MM, at level 2, to support the rule MM at level 1.

(b) Another way to try to justify MPP, which promises not to be vul-nerable to the difficulty that, if it is acceptable, so is an analogous jus-tification of MM, is suggested by Thomson’s discussion [1963] of theTortoise’s argument. Carroll’s tortoise, in [1895], refuses to draw theconclusion, ‘B,’ from ‘A ⊃ B’ and ‘A,’ insisting that a new premiss, ‘A⊃ ((A ⊃ B) ⊃ B)’ be added; and when that premiss is granted him, willstill not draw the conclusion, but insists on a further premiss, and soad indefinitum. Thomson argues that Achilles should never have con-ceded that an extra premiss was needed; for, he argues, if the originalinference was valid (semantically) the added premiss is true but notneeded, and if the original inference was invalid (semantically) theadded premiss is needed but false. There is an analogy, here, again,with attempts to justify induction by appending a premiss—something,usually, to the effect that ‘Nature is uniform’—which turns inferencesin accordance with RI into deductively valid inferences. The requiredpremiss would, presumably, be true but not needed if RI were deduc-tively valid, false but needed if it is not.

Thomson’s idea suggests that we should contrast this picture in thecase of MPP:

A5 (1) A ⊃ ((A ⊃ B) ⊃ B) (true but superfluous premiss)(2) A assumption(3) (A ⊃ B) ⊃ B 1, 2, MPP(4) A ⊃ B assumption(5) B 3, 4, MPP

with this picture in the case of MM:

156 SUSAN HAACK


A6 (1) (B ⊃ (A ⊃ B) ⊃ A) (false but needed premiss)(2) (A ⊃ B) ⊃ A assumption(3) B 1, 2, MM(4) A ⊃ B assumption(5) A 3, 4, MM

Thomson’s point is that in A5 premiss (1) is a tautology, so true; but itis not needed, since lines (2), (4) and (5) alone constitute a valid argu-ment. In A6, by contrast, premiss (1) is not a tautology; but it is needed,because lines (2), (4) and (5) alone do not constitute a valid argument.But this is to assume that MPP, which is the rule of inference in virtueof which in A5 (2) and (4) yield (5), is valid; whereas MM, which isthe rule of inference in virtue of which, in A6, (2) and (4) would yield(5), is not valid. But this is just what was to have been shown.

If A5 justifies MPP, which, after all, it uses, then the following argu-ment equally justifies MM:

A7 (1) (A ⊃ B) ⊃ (A ⊃ B) (true but superfluous premiss)(2) A ⊃ B assumption(3) A ⊃ B 1, 2, MM(4) B assumption(5) A 3, 4, MM

In A7 as in A5 the first premiss is a tautology, so true, but it is super-fluous, since (if MM is accepted) lines (2), (4) and (5) alone constitutea valid argument.

(c) Nor will it do to argue that MPP is, whereas MM is not, justified‘in virtue of the meaning of “⊃”’. For how is the meaning of ‘⊃’ given?There are three kinds of answer commonly given: that the meaning ofthe connectives is given by the rules of inference/axioms of the system



in which they occur; that the meaning is given by the interpretation, or,specifically, the truth-table, provided; that the meaning is given by theEnglish readings of the connectives. Well, if ‘⊃’ is supposed to be atleast partially defined by the rules of inference governing sentencescontaining it (cf. Prior [1960], [1964]) then MPP and MM would beexactly on a par. In a system containing MPP the meaning of ‘⊃’ is par-tially defined by the rule, from ‘A ⊃ B’ and ‘A,’ to infer ‘B.’ In a systemcontaining MM the meaning of ‘⊃’ is partially defined by the rule, from‘A ⊃ B’ and ‘B’ to infer ‘A.’ In either case the rule in question wouldbe justified in virtue of the meaning of ‘⊃,’ finally, since the meaningof ‘⊃’ would be given by the rule. If, on the other hand, we thought of‘⊃’ as partially defined by its truth-table (cf. Stevenson [1961]), we arein the difficulty discussed earlier ((a) above) that arguments from thetruth-table to the justification of a rule of inference are liable to employthe rule in question. Nor would it do to appeal to the usual reading of‘⊃’ as ‘if . . . then . . .,’ not just because the propriety of that readinghas been doubted, but also because the question, why ‘B’ follows from‘if A then B’ and ‘A’ but not ‘A’ from ‘if A then B’ and ‘B,’ is preciselyanalogous to the question at issue.

(d) Our arguments against attempted justifications of MPP have ap-pealed to the fact that analogous procedures would justify MM. So atthis point it might be suggested that we can produce independent argu-ments against MM. (Compare attempts to diagnose incoherence inRCI.) In particular, it might be supposed that it is a relatively simplematter to show that MM cannot be truth-preserving, since with MM atour disposal we could argue as follows:

A8 (1) (p & ~ p) ⊃ (pv ~ p)(2) pv ~ p(3) p & ~ p 1, 2 MM

158 SUSAN HAACK


So that a system including MM would be inconsistent. (This idea issuggested by Belnap’s paper on ‘tonk.’)

However, this argument is inconclusive because it depends upon cer-tain assumptions about what else we have in the system to which MMis appended—in particular, that (1) and (2) are theorems. Now certainlyif a system contained (1) and (2) as theorems, then (3) could be derivedby MM, and the system would be inconsistent; but a system allowingMM can hardly be assumed to be otherwise conventional. (After all,many systems lack ‘pv ~ p’ as a theorem; and minimal logic also lacks‘p ⊃ (~ p ⊃ q)’.)

(5) It might be suggested at this point that to direct our search forjustification to a form of argument, or argument schema, such as MPP,is misguided, that the justification of the schema lies in the validity ofits instances. So the answer to the question, ‘What justifies the conclu-sion?’ is simply ‘The premisses’; and the answer to the further question,what justifies the argument schema, is simply that its instances arevalid.

This suggestion is unsatisfactory for several reasons. First, it shiftsthe justification problem from the argument schema to its instances,without providing any solution to the problem of the justification of theinstances, beyond the bald assertion that they are justified. The claimthat one can just see that the premisses justify the conclusion is implau-sible in the extreme in view of the fact that people can and do disagreeabout which arguments are valid. Second, there is an implicit generalityin the claim that a particular argument is valid. For to say that an argu-ment is valid is not just to say that its premisses and its conclusion aretrue—for that is neither necessary nor sufficient for (semantic) validity.Rather, it is to say that its premisses could not be true without its con-clusion being true also, i.e. that there is no argument of that form withtrue premisses and false conclusion. But if the claim that a particular



argument is valid is to be spelled out by appeal to other arguments ofthat form, it is hopeless to try to justify that form of argument by appealto the validity of its instances. (Indeed, it is not a simple matter to specifyof what schema a particular argument is an instance. Our decision aboutwhat the logical form of an argument is may depend upon our viewabout whether the argument is valid.) Third, since a valid schema hasinfinitely many instances, if the validity of the schema were to be provenon the basis of the validity of its instances, the justification of the schemawould have to be inductive, and would in consequence inevitably failto establish a result of the desired strength. (Cf. Section 1.)

In rejecting this suggestion I do not, of course, deny the geneticpoint, that the codification of valid forms of inference, the constructionof a formal system, may proceed in part via generalisation over cases—though in part, I think, the procedure may also go in the opposite di-rection. (This genetic point is, I think, related to the one Carnap [1968]is making when he observes that we could not convince a man who is‘deductively blind’ of the validity of MPP.) But I do claim that the jus-tification of a form of inference cannot derive from intuition of the va-lidity of its instances.

(6) What I have said in this paper should, perhaps, be already famil-iar—it foreshadowed in Carroll [1895], and more or less explicit inQuine [1936] and Carnap 1968 (‘. . . the epistemological situation ininductive logic . . . is not worse than that in deductive logic, but quiteanalogous to it’, p. 266). But the point does not seem to have beentaken.

The moral of the paper might be put, pessimistically, as that deduc-tion is no less in need of justification than induction; or, optimistically,as that induction is in no more need of justification than deduction. Buthowever we put it, the presumption, that induction is shaky but deduc-

160 SUSAN HAACK


tion is firm, is impugned. And this presumption is quite crucial, e.g. toPopper’s proposal [1959] to replace inductivism by deductivism. Thoseof us who are sceptical about the analytic/synthetic distinction will, nodoubt, find these consequences less unpalatable than will those whoaccept it. And those of us who take a tolerant attitude to nonstandardlogics—who regard logic as a theory, revisable, like other theories, inthe light of experience—may even find these consequences welcome.



14

The Problem of

Counterfactual Conditionals1

Nelson Goodman

I. THE PROBLEM IN GENERAL

The analysis of counterfactual conditionals is no fussy little grammat-ical exercise. Indeed, if we lack the means for interpreting counterfac-tual conditionals, we can hardly claim to have any adequate philosophyof science. A satisfactory definition of scientific law, a satisfactory the-ory of confirmation or of disposition terms (and this includes not onlypredicates ending in “ible” and “able” but almost every objective pred-icate, such as “is red”), would solve a large part of the problem of coun-terfactuals. Accordingly, the lack of a solution to this problem impliesthat we have no adequate treatment of any of these other topics. Con-versely, a solution to the problem of counterfactuals would give us theanswer to critical questions about law, confirmation, and the meaningof potentiality.

I am not at all contending that the problem of counterfactuals is log-ically or psychologically the first of these related problems. It makeslittle difference where we start if we can go ahead. If the study of coun-terfactuals has up to now failed this pragmatic test, the alternative ap-proaches are little better off.

163


What, then, is the problem about counterfactual conditionals? Letus confine ourselves to those in which antecedent and consequent areinalterably false—as, for example, when I say of a piece of butter thatwas eaten yesterday, and that had never been heated,

If that piece of butter had been heated to 150° F., it would havemelted.

Considered as truth-functional compounds, all counterfactuals are ofcourse true, since their antecedents are false. Hence

If that piece of butter had been heated to 150° F., it would nothave melted

would also hold. Obviously something different is intended, and theproblem is to define the circumstances under which a given counter-factual holds while the opposing conditional with the contradictory con-sequent fails to hold. And this criterion of truth must be set up in theface of the fact that a counterfactual by its nature can never be subjectedto any direct empirical test by realizing its antecedent.

In one sense the name “problem of counterfactuals” is misleading,because the problem is independent of the form in which a given state-ment happens to be expressed. The problem of counterfactuals isequally a problem of factual conditionals, for any counterfactual canbe transposed into a conditional with a true antecedent and consequent;e.g.,

Since that butter did not melt, it wasn’t heated to 150° F.

The possibility of such transformation is of no great importance exceptto clarify the nature of our problem. That “since’’ occurs in the contra-

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positive shows that what is in question is a certain kind of connectionbetween the two component sentences; and the truth of this kind ofstatement—whether it is in the form of a counterfactual or factual con-ditional or some other form—depends not upon the truth or falsity ofthe components but upon whether the intended connection obtains.Recognizing the possibility of transformation serves mainly to focusattention on the central problem and to discourage speculation as to thenature of counterfacts. Although I shall begin my study by consideringcounterfactuals as such, it must be borne in mind that a general solutionwould explain the kind of connection involved irrespective of any as-sumption as to the truth or falsity of the components.

The effect of transposition upon another kind of conditional, whichI call “semifactual,’’ is worth noticing briefly. Should we assert

Even if the match had been scratched, it still would not havelighted,

we would uncompromisingly reject as an equally good expression ofour meaning the contrapositive,

Even if the match lighted, it still wasn’t scratched.

Our original intention was to affirm not that the non-lighting could beinferred from the scratching, but simply that the lighting could not beinferred from the scratching. Ordinarily a semifactual conditional hasthe force of denying what is affirmed by the opposite, fully counterfac-tual conditional. The sentence

Even had that match been scratched, it still wouldn’t have lighted

is normally meant as the direct negation of

14 The Problem of Counterfactual Conditionals 165


Had the match been scratched, it would have lighted.

That is to say, in practice full counterfactuals affirm, while semifactualsdeny, that a certain connection obtains between antecedent and conse-quent.2 Thus it is clear why a semifactual generally has not the samemeaning as its contrapositive.

There are various special kinds of counterfactuals that present spe-cial problems. An example is the ease of “counteridenticals,” illustratedby the statements

If I were Julius Caesar, I wouldn’t be alive in the twentiethcentury,

and

If Julius Caesar were I, he would be alive in the twentiethcentury.

Here, although the antecedent in the two cases is a statement of thesame identity, we attach two different consequents which, on the veryassumption of that identity, are incompatible. Another special class ofcounterfactuals is that of the “countercomparatives,” with antecedentssuch as

If I had more money, . . .

The trouble with these is that when we try to translate the counterfactualinto a statement about a relation between two tenseless, non-modal sen-tences, we get as an antecedent something like

If “I have more money than I have” were true, . . .

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although use of a self-contradictory antecedent was plainly not the orig-inal intent. Again there are the “counterlegals,” conditionals with an-tecedents that either deny general laws directly, as in

If triangles were squares, . . .

or else make a supposition of particular fact that is not merely false butimpossible, as in

If this cube of sugar were also spherical, . . .

All these kinds of counterfactuals offer interesting but not insur-mountable special difficulties.3 In order to concentrate upon the majorproblems concerning counterfactuals in general, I shall usuallychoose my examples in such a way as to avoid these more specialcomplications.

As I see it, there are two major problems, though they are not inde-pendent and may even be regarded as aspects of a single problem. Acounterfactual is true if a certain connection obtains between the an-tecedent and the consequent. But as is obvious from examples alreadygiven, the consequent seldom follows from the antecedent by logicalone. (1) In the first place, the assertion that a connection holds is madeon the presumption that certain circumstances not stated in the an-tecedent obtain. When we say

If that match had been scratched, it would have lighted,

we mean that conditions are such—i.e., the match is well made, is dryenough, oxygen enough is present, etc.—that “That match lights” canbe inferred from “That match is scratched.” Thus the connection weaffirm may be regarded as joining the consequent with the conjunction



of the antecedent and other statements that truly describe relevantconditions. Notice especially that our assertion of the counterfactualis not conditioned upon these circumstances obtaining. We do not as-sert that the counterfactual is true if the circumstances obtain; rather,in asserting the counterfactual we commit ourselves to the actual truthof the statements describing the requisite relevant conditions. Thefirst major problem is to define relevant conditions; to specify whatsentences are meant to be taken in conjunction with an antecedent asa basis for inferring the consequent. (2) But even after the particularrelevant conditions are specified, the connection obtaining will notordinarily be a logical one. The principle that permits inference of

That match lights

from

That match is scratched. That match is dry enough. Enoughoxygen is present. Etc.

is not a law of logic but what we call a natural or physical or causallaw. The second major problem concerns the definition of such laws.

II. THE PROBLEM OF RELEVANT CONDITIONS

It might seem natural to propose that the consequent follows by lawfrom the antecedent and a description of the actual state-of-affairs ofthe world, that we need hardly define relevant conditions because itwill do no harm to include irrelevant ones. But if we say that the con-sequent follows by law from the antecedent and all true statements, weencounter an immediate difficulty:—among true sentences is the negateof the antecedent, so that from the antecedent and all true sentences

168 NELSON GOODMAN


everything follows. Certainly this gives us no way of distinguishingtrue from false counterfactuals.

We are plainly no better off if we say that the consequent must followfrom some set of true statements conjoined with the antecedent; for givenany counterfactual antecedent A, there will always be a set S—namely,the set consisting of – A—such that from A&S any consequent follows.(Hereafter I shall regularly use “A” for the antecedent, “C” for the con-sequent, and “S” for the set of statements of the relevant conditions.)

Perhaps then we must exclude statements logically incompatiblewith the antecedent. But this is insufficient; for a parallel difficultyarises with respect to true statements which are not logically but areotherwise incompatible with the antecedent. For example, take

If that radiator had frozen, it would have broken.

Among true sentences may well be (S)

That radiator never reached a temperature below 33° F.

Now it is certainly generally true that

All radiators that freeze but never reach below 33° F. break,

and also that

All radiators that freeze but never reach below 33° F. fail to break;

for there are no such radiators. Thus from the antecedent of the coun-terfactual and the given S, we can infer any consequent.

The natural proposal to remedy this difficulty is to rule that coun-terfactuals can not depend upon empty laws; that the connection can



be established only by a principle of the form “All x’s are y’s” whenthere are some x’s. But this is ineffectual. For if empty principles areexcluded, the following non-empty principles may be used in the casegiven with the same result:

Everything that is either a radiator that freezes but does not reachbelow 33° F., or that is a soap bubble, breaks;Everything that is either a radiator that freezes but does not reachbelow 33° F., or is powder, does not break.

By these principles we can infer any consequent from the A and S inquestion.

The only course left open to us seems to be to define relevant con-ditions as the set of all true statements each of which is both logicallyand non-logically compatible with A where non-logical incompatibilitymeans violation of a non-logical law.4 But another difficulty immedi-ately appears. In a counterfactual beginning

If Jones were in Carolina, . . .

the antecedent is entirely compatible with

Jones is not in South Carolina

and with

Jones is not in North Carolina

and with

North Carolina plus South Carolina is identical with Carolina;

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but all these taken together with the antecedent make a set that is self-incompatible, so that again any consequent would be forthcoming.

Clearly it will not help to require only that for some set S of true sen-tences, A&S be self-compatible and lead by law to the consequent ; forthis would make a true counterfactual of

If Jones were in Carolina, he would be in South Carolina,

and also of

If Jones were in Carolina, he would be in North Carolina,

which can not both be true.It seems that we must elaborate our criterion still further, to charac-

terize a counterfactual as true if and only if there is some set S of truestatements such that A&S is self-compatible and leads by law to the con-sequent, while there is no such set S' such that A&S' is self-compatibleand leads by law to the negate of the consequent.5 Unfortunately eventhis is not enough. For among true sentences will be the negate of theconsequent: ~ C. Is ~ C compatible with A or not? If not, then A alonewithout any additional conditions must lead by law to C. But if ~ C iscompatible with A (as in most cases), then if we take ~ C as our S, theconjunction A&S will give us ~ C. Thus the criterion we have set upwill seldom be satisfied; for since ~ C will normally be compatible withA—as the need for introducing the relevant conditions testifies—therewill normally be an A (namely, ~ C) such that A&S is self-compatibleand leads by law to ~ C.

Part of our trouble lies in taking too narrow a view of our problem.We have been trying to lay down conditions under which an A thatis known to be false leads to a C that is known to be false; but it isequally important to make sure that our criterion does not establish



a similar connection between our A and the (true) negate of C. Be-cause our S together with A was to be so chosen as to give us C, itseemed gratuitous to specify that S must be compatible with C; andbecause ~ C is true by supposition, S would necessarily be compat-ible with it. But we are testing whether our criterion not only admitsthe true counterfactual we are concerned with but also excludes theopposing conditional. Accordingly, our criterion must be modifiedby specifying that S be compatible with both C and ~ C.6 In otherwords, S by itself must not decide between C and ~ C, but S togetherwith A must lead to C but not to ~ C. We need not know whether Cis true or false.

Our rule thus reads that a counterfactual is true if and only if thereis some set S of true sentences such that S is compatible with C andwith ~ C, and such that A&S is self-compatible and leads by law to C;while there is no set S' compatible with C and with ~ C, and such thatA&S' is self-compatible and leads by law to ~ C. As thus stated, therule involves a certain redundancy; but simplification is not in pointhere, for the criterion is still inadequate.

The requirement that A&S be self-compatible is not strong enough;for S might comprise true sentences that although compatible with A,were such that they would not be true if A were true. For this reason,many statements that we would regard as definitely false would be trueaccording to the stated criterion. As an example, consider the familiarcase where for a given match M, we would affirm

(I) If match M had been scratched, it would have lighted,

but deny

(II) If match M had been scratched, it would not have been dry.7

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According to our tentative criterion, statement II would be quite as trueas statement I. For in the case of II, we may take as an element in ourS the true sentence

Match M did not light,

which is presumably compatible with A (otherwise nothing would berequired along with A to reach the opposite as the consequent of thetrue counterfactual statement, I). As our total A&S we may have

Match M is scratched. It does not light. It is well made. Oxygenenough is present . . . etc.;

and from this, by means of a legitimate general law, we can infer

It was not dry

and there would seem to be no suitable set of sentences S' such thatA&S' leads by law to the negate of this consequent. Hence the unwantedcounterfactual is established in accord with our rule. The trouble iscaused by including in our S a true statement which though compatiblewith A would not be true if A were. Accordingly we must exclude suchstatements from the set of relevant conditions; S, in addition to satisfy-ing the other requirements already laid down, must be not merely com-patible with A but “jointly tenable” or “cotenable” with A. A iscotenable with S, and the conjunction A&S self-cotenable, if it is notthe case that S would not be true if A were.8

Parenthetically it may be noted that the relative fixity of conditionsis often unclear, so that the speaker or writer has to make explicit ad-ditional provisos or give subtle verbal clues as to his meaning. For



example, each of the following two counterfactuals would normally beaccepted:

If New York City were in Georgia, then New York City would bein the South. If Georgia included New York City, then Georgiawould not be entirely in the South.

Yet the antecedents are logically indistinguishable. What happens isthat the direction of expression becomes important, because in the for-mer case the meaning is

If New York City were in Georgia, and the boundaries of Georgiaremained unchanged, then . . .

while in the latter case the meaning is

If Georgia included New York City, and the boundaries of NewYork City remained unchanged, then . . .

Without some such cue to the meaning as is covertly given by the word-order, we should be quite uncertain which of the two consequents inquestion could be truly attached. The same kind of explanation ac-counts for the paradoxical pairs of counteridenticals mentioned earlier.

Returning now to the proposed rule, I shall neither offer further cor-rections of detail nor discuss whether the requirement that S be coten-able with A makes superfluous some other provisions of the criterion;for such matters become rather unimportant beside the really seriousdifficulty that now confronts us. In order to determine the truth of agiven counterfactual it seems that we have to determine, among other

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things, whether there is a suitable S that is cotenable with A and meetscertain further requirements. But in order to determine whether or nota given S is cotenable with A, we have to determine whether or not thecounterfactual “If A were true, then S would not be true” is itself true.But this means determining whether or not there is a suitable S1, coten-able with A, that leads to ~ S and so on. Thus we find ourselves involvedin an infinite regressus or a circle; for cotenability is defined in termsof counterfactuals, yet the meaning of counterfactuals is defined interms of cotenability. In other words to establish any counterfactual, itseems that we first have to determine the truth of another. If so, we cannever explain a counterfactual except in terms of others, so that theproblem of counterfactuals must remain unsolved.

Though unwilling to accept this conclusion, I do not at present seeany way of meeting the difficulty. One naturally thinks of revising thewhole treatment of counterfactuals in such a way as to admit first thosethat depend on no conditions other than the antecedent, and then usethese counterfactuals as the criteria for the cotenability of relevant con-ditions with antecedents of other counterfactuals, and so on. But thisidea seems initially rather unpromising in view of the formidable dif-ficulties of accounting by such a step-by-step method for even so sim-ple a counterfactual as :

If the match had been scratched, it would have lighted.

III. THE PROBLEM OF LAW

Even more serious is the second of the problems mentioned earlier: thenature of the general statements that enable us to infer the consequentupon the basis of the antecedent and the statement of relevant condi-tions. The distinction between these connecting principles and relevant



conditions is imprecise and arbitrary; the “connecting principles” mightbe conjoined to the condition-statements, and the relation of the an-tecedent-conjunction (A&S) to the consequent thus made a matter oflogic. But the same problems would arise as to the kind of principlethat is capable of supporting a counterfactual; and it is convenient toconsider the connecting principles separately.

In order to infer the consequent of a counterfactual from the an-tecedent A and a suitable statement of relevant conditions S, we makeuse of a general statement, namely, the generalization9 of the condi-tional having A&S for antecedent and C for consequent. For example,in the case of

If the match had been scratched, it would have lighted

the connecting principle is

Every match that is scratched, well made, dry enough, in enoughoxygen, etc., lights.

But notice that not every counterfactual is actually supported by theprinciple thus arrived at, even if that principle is true. Suppose, for ex-ample, that all I had in my right pocket on V–E day was a group of sil-ver coins. Now we would not under normal circumstances affirm of agiven penny P

If P had been in my pocket on V–E day, P would have beensilver,10

even though from

P was in my pocket on V–E day

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we can infer the consequent by means of the general statement

Everything in my pocket on V–E day was silver.

On the contrary, we would assert that if P had been in my pocket, thenthis general statement would not be true. The general statement willnot permit us to infer the given consequent from the counterfactual as-sumption that P was in my pocket, because the general statement willnot itself withstand that counterfactual assumption. Though the sup-posed connecting principle is indeed general, true, and perhaps evenfully confirmed by observation of all cases, it is incapable of supportinga counterfactual because it remains a description of accidental fact, nota law. The truth of a counterfactual conditional thus seems to dependon whether the general sentence required for the inference is a law ornot. If so, our problem is to distinguish accurately between causal lawsand casual facts.11

The problem illustrated by the example of the coins is closely relatedto that which led us earlier to require the cotenability of the antecedentand the relevant conditions, in order to avoid resting a counterfactualon any statement that would not be true if the antecedent were true. Butdecision as to the cotenability of two sentences must depend upon de-cisions as to whether or not certain general statements are laws, andwe are now concerned directly with the latter problem. Is there someway of distinguishing laws from non-laws among true universal state-ments of the kind in question, such that a law will be the sort of prin-ciple that will support a counterfactual conditional while a non-law willnot?

Any attempt to draw the distinction by reference to a notion ofcausative force can be dismissed at once as unscientific. And it is clearthat no purely syntactical criterion can be adequate, for even the mostspecial descriptions of particular facts can be cast in a form having any



desired degree of syntactical universality. “Book B is small” becomes“Everything that is Q is small” if “Q” stands for some predicate thatapplies uniquely to B. What then does distinguish a law like

All butter melts at 150° F.

from a true and general non-law like

All the coins in my pocket are silver?

Primarily, I would like to suggest, the fact that the first is accepted astrue while many cases of it remain to be determined, the further, unex-amined cases being predicted to conform with it. The second sentence,on the contrary, is accepted as a description of contingent fact after thedetermination of all cases, no prediction of any of its instances beingbased upon it. This proposal raises innumerable problems, some ofwhich I shall consider presently; but the idea behind it is just that theprinciple we use to decide counterfactual cases is a principle we arewilling to commit ourselves to in deciding unrealized cases that are stillsubject to direct observation.

As a first approximation then, we might say that a law is a true sen-tence used for making predictions. That laws are used predictively isof course a simple truism, and I am not proposing it as a novelty. I wantonly to emphasize the idea that rather than a sentence being used forprediction because it is a law, it is called a law because it is used forprediction; and that rather than the law being used for prediction be-cause it describes a causal connection, the meaning of the causal con-nection is to be interpreted in terms of predictively used laws.

By the determination of all instances, I mean simply the examinationor testing by other means of all things that satisfy the antecedent, to

178 NELSON GOODMAN


decide whether all satisfy the consequent also. There are difficult ques-tions about the meaning of “instance,” many of which ProfessorHempel has investigated. Most of these are avoided in our present studyby the fact that we are concerned with a very narrow class of sentences:those arrived at by generalizing conditionals of a certain kind. Remain-ing problems about the meaning of “instance” I shall have to ignorehere. As for “determination,” I do not mean final discovery of truth,but only enough examination to reach a decision as to whether a givenstatement or its negate is to be admitted as evidence for the hypothesisin question.

The limited scope of our present problem makes it unimportant thatour criterion, if applied generally to all statements, would classify aslaws many statements—e.g., true singular predictions— that we wouldnot normally call laws.

A more pertinent point is the application of the proposed criterionto vacuous generalities. As the criterion stands, no conditional with anempty antecedent-class will be a law, for all its instances will have beendetermined prior to its acceptance. Now since the antecedents of thestatements we are concerned with will be generalizations from self-cotenable and therefore self-compatible conjunctions, none will beknown to be vacuous.12 For example, since

M is scratched. M is dry . . . (etc.)

is a self-compatible set, the antecedent of

For every x, if x is scratched and x is dry (etc.), then x lights

will not be known to be false. But now we would still want the gener-alized principle just given to be a law if it should just happen to be the



case that nothing satisfies the antecedent. This discloses a defect in ourcriterion, which should be amended to read as follows: A true statementof the kind in question is a law if we accept it before we know that theinstances we have determined are all the instances.

For convenience, I shall use the term “lawlike” for sentences which,whether they are true or not, satisfy the other requirements in the defi-nition of law. A law is thus a sentence that is both lawlike and true, buta sentence may be true without being lawlike, as I have illustrated, orlawlike without being true, as we are always learning to our dismay.

Now the property of lawlikeness as so far defined is not only ratheran accidental and subjective one but an ephemeral one that sentencesmay acquire and lose. As an example of the undesirable consequencesof this impermanence, a true sentence that had been used predictivelywould cease to be a law when it became fully tested—i.e., when noneof its instances remained undetermined. The definition, then, must berestated in some such way as this: A general statement is lawlike if andonly if it is acceptable prior to the determination of all its instances.This is immediately objectionable because “acceptable” itself is plainlya dispositional term; but I propose to use it only tentatively, with theidea of eliminating it eventually by means of a non-dispositional defi-nition. Before trying to accomplish that, however, we must face anotherdifficulty in our tentative criterion of lawlikeness.

Suppose that the appropriate generalization fails to support a givencounterfactual because that generalization, while true, is un-lawlike, asis

Everything in my pocket is silver.

All we would need do to get a law would be to broaden the antecedentstrategically. Consider, for example, the sentence

180 NELSON GOODMAN


Everything that is in my pocket or is a dime is silver.

Since we have not examined all dimes, this is a predictive statementand—since presumably true—would be a law. Now if we consider ouroriginal counterfactual and choose our S so that A&S is

P is in my pocket. P is in my pocket or is a dime,

then the pseudo-law just constructed can be used to infer from this thesentence “P is silver.” Thus the untrue counterfactual is established, ifone prefers to avoid an alternation as a condition-statement; the sameresult can be obtained by using a new predicate such as “dimo” to mean“is in my pocket or is a dime.”13

The change called for, I think, will make the definition of law-likenessread as follows: A sentence is lawlike if its acceptance does not dependupon the determination of any given instance.14 Naturally this does notmean that acceptance is to be independent of all determination of in-stances, but only that there is no particular instance on the determina-tion of which acceptance depends. This criterion excludes from theclass of laws a statement like

That book is black and oranges are spherical

on the ground that acceptance requires knowing whether the book isblack; it excludes

Everything that is in my pocket or is a dime is silver

on the ground that acceptance demands examination of all things in mypocket. Moreover, it excludes a statement like



All the marbles in this bag except Number 19 are red, andNumber 19 is black

on the ground that acceptance would depend on examination of orknowledge gained otherwise concerning marble Number 19. In fact theprinciple involved in the proposed criterion is a rather powerful oneand seems to exclude most of the troublesome cases.

We must still, however, replace the notion of the acceptability of asentence, or of its acceptance depending or not depending on some givenknowledge, by a positive definition of such dependence. It is clear thatto say that the acceptance of a given statement depends upon a certainkind and amount of evidence is to say that given such evidence, accept-ance of the statement is in accord with certain general standards for theacceptance of statements that are not fully tested. So one turns naturallyto theories of induction and confirmation to learn the distinguishing fac-tors or circumstances that determine whether or not a sentence is ac-ceptable without complete evidence. But publications on confirmationnot only have failed to make clear the distinction between confirmableand non-confirmable statements, but show little recognition that such aproblem exists.15 Yet obviously in the case of some sentences like

Everything in my pocket is silver

or

No twentieth-century president of the United States will bebetween 6 feet 1 inch and 6 feet 1½ inches tall,

not even the testing with positive results of all but a single instance islikely to lead us to accept the sentence and predict that the one remain-ing instance will conform to it; while for other sentences such as

182 NELSON GOODMAN


All dimes are silver

or

All butter melts at 150° F.

or

All flowers of plants descended from this seed will be yellow

positive determination of even a few instances may lead us to acceptthe sentence with confidence and make predictions in accordance withit.

There is some hope that cases like these can be dealt with by a suffi-ciently careful and intricate elaboration of current confirmation theories;but inattention to the problem of distinguishing between confirmableand non-confirmable sentences has left most confirmation theories opento more damaging counterexamples of an elementary kind.

Suppose we designate the 26 marbles in a sack by the letters of thealphabet, using these merely as proper names having no ordinal sig-nificance. Suppose further that we are told that all the marbles exceptd are red, but we are not told what color d is. By the usual kind of con-firmation theory this gives strong confirmation for the statement

Ra. Rb. Rc. Rd. . . . Rz

because 25 of the 26 cases are known to be favorable while none isknown to be unfavorable. But unfortunately the same argument wouldshow that the very same evidence would equally confirm

Ra. Rb. Rc. Re. . . . Rz—Rd,



for again we have 25 favorable and no unfavorable cases. Thus “Rd”and “~ Rd” are equally and strongly confirmed by the same evidence.If I am required to use a single predicate instead of both “R” and “~ R”in the second case, I will use “P” to mean:

is in the sack and either is not d and is red, or is d and is not red.

Then the evidence will be 25 positive cases for

All the marbles are P

from which it follows that d is P, which implies that d is not red. Theproblem of what statements are confirmable merely becomes the equiv-alent problem of what predicates are projectible from known to un-known cases.

So far, I have discovered no way of meeting these difficulties. Yetas we have seen, some solution is urgently wanted for our present pur-pose; for only where willingness to accept a statement involves predic-tions of instances that may be tested does acceptance endow thatstatement with the authority to govern counterfactual cases, which cannot be directly tested.

In conclusion, then, some problems about counterfactuals dependupon the definition of cotenability, which in turn seems to depend uponthe prior solution of those problems. Other problems require an ade-quate definition of law. The tentative criterion of law here proposed isreasonably satisfactory in excluding unwanted kinds of statements, andin effect, reduces one aspect of our problem to the question how to de-fine the circumstances under which a statement is acceptable independ-ently of the determination of any given instance. But this question I donot know how to answer.

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NOTES

1. Slightly revised version of a paper read before the New York Philo-sophical Circle, May 11, 1946. My indebtedness in several matters to thework of C. I. Lewis and of C. H. Langford has seemed too obvious to callfor detailed mention.

2. The practical import of a semifactual is thus different from its literalmeaning. Literally a semifactual and the corresponding counterfactual arenot contradictories but contraries, and both may be false (cf. footnote 8). Thepresence of the auxiliary terms “even” and “still,” or either of them, is per-haps the idiomatic indication that a not quite literal meaning is intended.

3. Of the special kinds of counterfactuals mentioned, I shall have some-thing to say later about counteridenticals and counterlegals. As for counter-comparatives, the following procedure is appropriate:—Given “If I hadarrived one minute later, I would have missed the train,” first expand this to“(∃t). t is a time. I arrived (d) at t. If I had arrived one minute later than t, Iwould have missed the train.” The counterfactual conditional constitutingthe final clause of this conjunction can then be treated, within the quantifiedwhole, in the usual way. Translation into “If ‘I arrive one minute later thant’ were true, then ‘I miss the train’ would have been true” does not give us aself-contradictory component.

4. This of course raises very serious questions, which I shall come topresently, about the nature of non-logical law.

5. Note that the requirement that A&S be self-compatible can be fulfilledonly if the antecedent is self-compatible; hence the conditionals I have called“counterlegal” will all be false. This is convenient for our present purposeof investigating counterfactuals that are not counterlegals. If it later appearsdesirable to regard all or some counterlegals as true, special provisions maybe introduced.

6. It is natural to inquire whether for similar reasons we should stipulatethat S must be compatible with both A and ~ A, but this is unnecessary. Forif S is incompatible with ~ A, then A follows from S; therefore if S is com-patible with both C and ~ C, then A&S can not lead by law to one but notthe other. Hence no sentence incompatible with ~ A can satisfy the other re-quirements for a suitable S.



7. Of course, some sentences similar to II, referring to other matchesunder special conditions, may be true; but the objection to the proposed cri-terion is that it would commit us to many such statements that are patentlyfalse. I am indebted to Morton G. White for a suggestion concerning the ex-position of this point.

8. The double negative can not be eliminated here; for “. . . if S would betrue if A were” actually constitutes a stronger requirement. As we noted ear-lier (footnote 2), if two conditionals having the same counterfactual an-tecedent are such that the consequent of one is the negate of the consequentof the other, the conditionals are contraries and both may be false. This willbe the case, for example, if every otherwise suitable set of relevant conditionsthat in conjunction with the antecedent leads by law either to a given conse-quent or its negate leads also to the other.

9. The sense of “generalization” intended here is that explained by C. G.Hempel in “A Purely Syntactical Definition of Confirmation,” Journal ofSymbolic Logic, Vol. 8 (1943), pp. 122–143.

10. The antecedent in this example is intended to mean “If P, while re-maining distinct from the things that were in fact in my pocket on V–E day,had also been in my pocket then,” and not the quite different, counteriden-tical “If P had been identical with one of the things that were in my pocketon V–E day.” While the antecedents of most counterfactuals (as, again, ourfamiliar one about the match) are—literally speaking—open to both sorts ofinterpretation, ordinary usage normally calls for some explicit indicationwhen the counteridentical meaning is intended.

11. The importance of distinguishing laws from non-laws is too oftenoverlooked. If a clear distinction can be defined, it may serve not only thepurposes explained in the present paper but also many of those for whichthe increasingly dubious distinction between analytic and synthetic state-ments is ordinarily supposed to be needed.

12. Had it been sufficient in the preceding section to require only thatA&S be self-compatible, this requirement might now be eliminated in favorof the stipulation that the generalization of the conditional having A&S asantecedent and C as consequent should be non-vacuous; but this stipulationwould not guarantee the self-cotenability of A&S.

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13. Apart from the special class of connecting principles we are concernedwith, note that under the stated criterion of lawlikeness, any statement couldbe expanded into a lawlike one; for example: given “This book is black” wecould use the predictive sentence “This book is black and all oranges arespherical” to argue that the blackness of the book is the consequence of alaw.

14. So stated, the definition counts vacuous principles as laws. If we readinstead “given class of instances,” vacuous principles will be non-laws sincetheir acceptance depends upon examination of the null class of instances.For my present purposes the one formulation is as good as the other.

15. The points discussed in this and the following paragraph have beendealt with a little more fully in my “Query on Confirmation,” this Journal,Vol. XLIII (1946), pp. 383–385.



15

On What There Is1

Willard V. Quine

A curious thing about the ontological problem is its simplicity. It canbe put in three Anglo-Saxon monosyllables: “What is there?” It can beanswered, moreover, in a word—“Everything”—and everyone will ac-cept this answer as true. However, this is merely to say that there iswhat there is. There remains room for disagreement over cases; and sothe issue has stayed alive down the centuries.

Suppose now that two philosophers, McX and I, differ over ontol-ogy. Suppose McX maintains there is something which I maintain thereis not. McX can, quite consistently with his own point of view, describeour difference of opinion by saying that I refuse to recognize certainentities. I should protest, of course, that he is wrong in his formulationof our disagreement, for I maintain that there are no entities, of the kindwhich he alleges, for me to recognize; but my finding him wrong in hisformulation of our disagreement is unimportant, for I am committed toconsidering him wrong in his ontology anyway.

When I try to formulate our difference of opinion, on the other hand,I seem to be in a predicament. I cannot admit that there are some thingswhich McX countenances and I do not, for in admitting that there aresuch things I should be contradicting my own rejection of them.

189


It would appear, if this reasoning were sound, that in any ontologicaldispute the proponent of the negative side suffers the disadvantage ofnot being able to admit that his opponent disagrees with him.

This is the old Platonic riddle of non-being. Non-being must in somesense be, otherwise what is it that there is not? This tangled doctrinemight be nicknamed Plato’s beard; historically it has proved tough, fre-quently dulling the edge of Occam’s razor.

It is some such line of thought that leads philosophers like McX toimpute being where they might otherwise be quite content to recognizethat there is nothing. Thus, take Pegasus. If Pegasus were not, McX ar-gues, we should not be talking about anything when we use the word;therefore it would be nonsense to say even that Pegasus is not. Thinkingto show thus that the denial of Pegasus cannot be coherently main-tained, he concludes that Pegasus is.

McX cannot, indeed, quite persuade himself that any region ofspace-time, near or remote, contains a flying horse of flesh and blood.Pressed for further details on Pegasus, then, he says that Pegasus is anidea in men’s minds. Here, however, a confusion begins to be apparent.We may for the sake of argument concede that there is an entity, andeven a unique entity (though this is rather implausible), which is themental Pegasus-idea; but this mental entity is not what people are talk-ing about when they deny Pegasus.

McX never confuses the Parthenon with the Parthenon-idea. TheParthenon is physical; the Parthenon-idea is mental (according anywayto McX’s version of ideas, and I have no better to offer). The Parthenonis visible; the Parthenon-idea is invisible. We cannot easily imaginetwo things more unlike, and less liable to confusion, than the Parthenonand the Parthenon-idea. But when we shift from the Parthenon to Pe-gasus, the confusion sets in—for no other reason than that McX would

190 WILLARD V. QUINE


sooner be deceived by the crudest and most flagrant counterfeit thangrant the non-being of Pegasus.

The notion that Pegasus must be, because it would otherwise be non-sense to say even that Pegasus is not, has been seen to lead McX intoan elementary confusion. Subtler minds, taking the same precept astheir starting point, come out with theories of Pegasus which are lesspatently misguided than McX’s, and correspondingly more difficult toeradicate. One of these subtler minds is named, let us say, Wyman. Pe-gasus, Wyman maintains, has his being as an unactualized possible.When we say of Pegasus that there is no such thing, we are saying,more precisely, that Pegasus does not have the special attribute of ac-tuality. Saying that Pegasus is not actual is on a par, logically, with say-ing that the Parthenon is not red; in either case we are saying somethingabout an entity whose being is unquestioned.

Wyman, by the way, is one of those philosophers who have unitedin ruining the good old word “exist.” Despite his espousal of unactu-alized possibles, he limits the word “existence” to actuality—thus pre-serving an illusion of ontological agreement between himself and uswho repudiate the rest of his bloated universe. We have all been proneto say, in our common-sense usage of “exist,” that Pegasus does notexist, meaning simply that there is no such entity at all. If Pegasus ex-isted he would indeed be in space and time, but only because the word“Pegasus” has spatio-temporal connotations, and not because “exists”has spatio-temporal connotations. If spatio-temporal reference is lack-ing when we affirm the existence of the cube root of 27, this is simplybecause a cube root is not a spatio-temporal kind of thing, and notbecause we are being ambiguous in our use of “exist.” However,Wyman, in an ill-conceived effort to appear agreeable, genially grantsus the non-existence of Pegasus and then, contrary to what we meant

15 On What There Is 191


by non-existence of Pegasus, insists that Pegasus is. Existence is onething, he says, and subsistence is another. The only way I know ofcoping with this obfuscation of issues is to give Wyman the word“exist.” I’ll try not to use it again; I still have “is.” So much for lexi-cography; let’s get back to Wyman’s ontology.

Wyman’s overpopulated universe is in many ways unlovely. It of-fends the æsthetic sense of us who have a taste for desert landscapes,but this is not the worst of it. Wyman’s slum of possibles is a breedingground for disorderly elements. Take, for instance, the possible fat manin that doorway; and, again, the possible bald man in that doorway. Arethey the same possible man, or two possible men? How do we decide?How many possible men are there in that doorway? Are there morepossible thin ones than fat ones? How many of them are alike? Orwould their being alike make them one? Are no two possible thingsalike? Is this the same as saying that it is impossible for two things tobe alike? Or, finally, is the concept of identity simply inapplicable tounactualized possibles? But what sense can be found in talking of en-tities which cannot meaningfully be said to be identical with themselvesand distinct from one another? These elements are well nigh incorrigi-ble. By a Fregean therapy of individual concepts, some effort might bemade at rehabilitation; but I feel we’d do better simply to clearWyman’s slum and be done with it.

Possibility, along with the other modalities of necessity and impos-sibility and contingency, raises problems upon which I do not mean toimply that we should turn our backs. But we can at least limit modali-ties to whole statements. We may impose the adverb “possibly” upona statement as a whole, and we may well worry about the semanticalanalysis of such usage; but little real advance in such analysis is to behoped for in expanding our universe to include so-called possible en-tities. I suspect that the main motive for this expansion is simply the



old notion that Pegasus, e.g., must be because it would otherwise benonsense to say even that he is not.

Still, all the rank luxuriance of Wyman’s universe of possibles wouldseem to come to naught when we make a slight change in the exampleand speak not of Pegasus but of the round square cupola on BerkeleyCollege. If, unless Pegasus were, it would be nonsense to say that he isnot, then by the same token, unless the round square cupola on BerkeleyCollege were, it would be nonsense to say that it is not. But, unlike Pe-gasus, the round square cupola on Berkeley College cannot be admittedeven as an unactualized possible. Can we drive Wyman now to admit-ting also a realm of unactualizable impossibles? If so, a good many em-barrassing questions could be asked about them. We might hope evento trap Wyman in contradictions, by getting him to admit that certainof these entities are at once round and square. But the wily Wymanchooses the other horn of the dilemma and concedes that it is nonsenseto say that the round square cupola on Berkeley College is not. He saysthat the phrase “round square cupola” is meaningless.

Wyman was not the first to embrace this alternative. The doctrine ofthe meaninglessness of contradictions runs away back. The traditionsurvives, moreover, in writers such as Wittgenstein, who seem to sharenone of Wyman’s motivations. Still I wonder whether the first tempta-tion to such a doctrine may not have been substantially the motivationwhich we have observed in Wyman. Certainly the doctrine has no in-trinsic appeal; and it has led its devotees to such quixotic extremes asthat of challenging the method of proof by reductio ad absurdum—achallenge in which I seem to detect a quite striking reductio ad absur-dum eius ipsius.

Moreover, the doctrine of meaninglessness of contradictions has thesevere methodological drawback that it makes it impossible, in princi-ple, ever to devise an effective test of what is meaningful and what is



not. It would be forever impossible for us to devise systematic ways ofdeciding whether a string of signs made sense—even to us individually,let alone other people—or not. For it follows from a discovery in math-ematical logic, due to Church, that there can be no generally applicabletest of contradictoriness.

I have spoken disparagingly of Plato’s beard, and hinted that it istangled. I have dwelt at length on the inconveniences of putting up withit. It is time to think about taking steps.

Russell, in his theory of so-called singular descriptions, showedclearly how we might meaningfully use seeming names without sup-posing that the entities allegedly named be. The names to which Rus-sell’s theory directly applies are complex descriptive names such as“the author of Waverly,” “the present King of France,” “the roundsquare cupola on Berkeley College.” Russell analyzes such phrases sys-tematically as fragments of the whole sentences in which they occur.The sentence, “The author of Waverly was a poet,” e.g.' is explained asa whole as meaning “Someone (better: something) wrote Waverly andwas a poet, and nothing else wrote Waverly.” (The point of this addedclause is to affirm the uniqueness which is implicit in the word “the,”in “the author of Waverly.”) The sentence “The round square cupolaon Berkeley College is pink” is explained as “Something is round andsquare and is a cupola on Berkeley College and is pink, and nothingelse is round and square and a cupola on Berkeley College.”

The virtue of this analysis is that the seeming name, a descriptivephrase, is paraphrased in context as a so-called incomplete symbol. Nounified expression is offered as an analysis of the descriptive phrase,but the statement as a whole which was the context of that phrase stillgets its full quota of meaning—whether true or false.

The unanalyzed statement “The author of Waverly was a poet” con-tains a part, “the author of Waverly,” which is wrongly supposed by



McX and Wyman to demand objective reference in order to be mean-ingful at all. But in Russell’s translation, “Something wrote Waverlyand was a poet and nothing else wrote Waverly,” the burden of objectivereference which had been put upon the descriptive phrase is now takenover by words of the kind that logicians call bound variables, variablesof quantification: namely, words like “something,” “nothing,” “every-thing.” These words, far from purporting to be names specifically ofthe author of Waverly, do not purport to be names at all; they refer toentities generally, with a kind of studied ambiguity peculiar to them-selves. These quantificational words or bound variables are, of coursea basic part of language, and their meaningfulness, at least in context,is not to be challenged. But their meaningfulness in no way presupposesthere being either the author of Waverly or the round square cupola onBerkeley College or any other specifically preassigned objects.

Where descriptions are concerned, there is no longer any difficultyin affirming or denying being. “There is the author of Waverly” is ex-plained by Russell as meaning “Someone (or, more strictly, something)wrote Waverly and nothing else wrote Waverly.” “The author of Waverlyis not” is explained, correspondingly, as the alternation “Either eachthing failed to write Waverly or two or more things wrote Waverly.” Thisalternation is false, but meaningful; and it contains no expression pur-porting to designate the author of Waverly. The statement “The roundsquare cupola on Berkeley College is not” is analyzed in similar fashion.So the old notion that statements of non-being defeat themselves goesby the board. When a statement of being or non-being is analyzed byRussell’s theory of descriptions, it ceases to contain any expression whicheven purports to name the alleged entity whose being is in question, sothat the meaningfulness of the statement no longer can be thought topresuppose that there be such an entity. Now, what of “Pegasus”? Thisbeing a word rather than a descriptive phrase, Russell’s argument does



not immediately apply to it. However, it can easily be made to apply.We have only to rephrase “Pegasus” as a description, in any way thatseems adequately to single out our idea: say, “the winged horse thatwas captured by Bellerophon.” Substituting such a phrase for “Pega-sus,” we can then proceed to analyze the statement “Pegasus is,” or“Pegasus is not,” precisely on the analogy of Russell’s analysis of “Theauthor of Waverly is” and “The author of Waverly is not.”

In order thus to subsume a one-word name or alleged name such as“Pegasus” under Russell’s theory of description, we must, of course,be able first to translate the word into a description. But this is no realrestriction. If the notion of Pegasus had been so obscure or so basic aone that no pat translation into a descriptive phrase had offered itselfalong familiar lines, we could still have availed ourselves of the fol-lowing artificial and trivial-seeming device: we could have appealedto the ex hypothesi unanalyzable, irreducible attribute of being Pegasus,adopting, for its expression, the verb “is-Pegasus,” or “pegasizes.” Thenoun “Pegasus” itself could then be treated as derivative, and identifiedafter all with a description: “the thing that is-Pegasus,” “the thing thatpegasizes.”

If the importing of such a predicate as “pegasizes” seems to commitus to recognizing that there is a corresponding attribute, pegasizing, inPlato’s heaven or in the mind of men, well and good. Neither we norWyman nor McX have been contending, thus far, about the being ornon-being of universals, but rather about that of Pegasus. If in terms ofpegasizing we can interpret the noun “Pegasus” as a description subjectto Russell’s theory of descriptions, then we have disposed of the oldnotion that Pegasus cannot be said not to be without presupposing thatin some sense Pegasus is.

Our argument is now quite general. McX and Wyman supposed thatwe could not meaningfully affirm a statement of the form “So-and-so



is not,” with a simple or descriptive singular noun in place of “so-and-so,” unless so-and-so be. This supposition is now seen to be quite gen-erally groundless, since the singular noun in question can always beexpanded into a singular description, trivially or otherwise, and thenanalyzed out à la Russell.

We cannot conclude, however, that man is henceforth free of all on-tological commitments. We commit ourselves outright to an ontologycontaining numbers when we say there are prime numbers between1000 and 1010; we commit ourselves to an ontology containing cen-taurs when we say there are centaurs; and we commit ourselves to anontology containing Pegasus when we say Pegasus is. But we do notcommit ourselves to an ontology containing Pegasus or the author ofWaverly or the round square cupola on Berkeley College when we saythat Pegasus or the author of Waverly or the cupola in question is not.We need no longer labour under the delusion that the meaningfulnessof a statement containing a singular term presupposes an entity namedby the term. A singular term need not name to be significant.

An inkling of this might have dawned on Wyman and McX evenwithout benefit of Russell if they had only noticed—as so few of us do—that there is a gulf between meaning and naming even in the case of asingular term which is genuinely a name of an object. Frege’s examplewill serve: the phrase “Evening Star” names a certain large physical ob-ject of spherical form, which is hurtling through space some scores ofmillions of miles from here. The phrase “Morning Star” names thesame thing, as was probably first established by some observant Baby-lonian. But the two phrases cannot be regarded as having the samemeaning; otherwise that Babylonian could have dispensed with his ob-servations and contented himself with reflecting on the meanings of hiswords. The meanings, then, being different from one another, must beother than the named object, which is one and the same in both cases.



Confusion of meaning with naming not only made McX think hecould not meaningfully repudiate Pegasus; a continuing confusion ofmeaning with naming no doubt helped engender his absurd notion thatPegasus is an idea, a mental entity. The structure of his confusion is asfollows. He confused the alleged named object Pegasus with the mean-ing of the word “Pegasus,” therefore concluding that Pegasus must bein order that the word have meaning. But what sorts of things are mean-ings? This is a moot point; however, one might quite plausibly explainmeanings as ideas in the mind, supposing we can make clear sense inturn of the idea of ideas in the mind. Therefore Pegasus, initially con-fused with a meaning, ends up as an idea in the mind. It is the more re-markable that Wyman, subject to the same initial motivation as McX,should have avoided this particular blunder and wound up with unac-tualized possibles instead.

Now let us turn to the ontological problem of universals: the ques-tion whether there are such entities as attributes, relations, classes,numbers, functions. McX, characteristically enough, thinks there are.Speaking of attributes, he says: “There are red houses, red roses, redsunsets; this much is prephilosophical common-sense in which wemust all agree. These houses, roses and sunsets, then, have somethingin common; and this which they have in common is all I mean by theattribute of redness.” For McX, thus, there being attributes is evenmore obvious and trivial than the obvious and trivial fact of there beingred houses, roses and sunsets. This, I think, is characteristic of meta-physics, or at least of that part of metaphysics called ontology: onewho regards a statement on this subject as true at all must regard it astrivially true. One’s ontology is basic to the conceptual scheme bywhich he interprets all experiences, even the most commonplace ones.Judged within some particular conceptual scheme—and how else isjudgment possible?—an ontological statement goes without saying,standing in need of no separate justification at all. Ontological state-



ments follow immediately from all manner of casual statements ofcommonplace fact, just as—from the point of view, anyway, of McX’sconceptual scheme—“There is an attribute” follows from “There arered houses, red roses, red sunsets.”

Judged in another conceptual scheme, an ontological statementwhich is axiomatic to McX’s mind may, with equal immediacy andtriviality, be adjudged false. One may admit that there are red houses,roses, and sunsets, but deny, except as a popular and misleading mannerof speaking, that they have anything in common. The words “houses,”“roses” and “sunsets” denote each of sundry individual entities whichare houses and roses and sunsets, and the word “red” or “red object”denotes each of sundry individual entities which are red houses, redroses, red sunsets; but there is not, in addition, any entity whatever, in-dividual or otherwise, which is named by the word “redness,” nor, forthat matter, by the word “househood,” “rosehood,” “sunsethood.” Thatthe houses and roses and sunsets are all of them red may be taken asultimate and irreducible, and it may be held that McX is no better off,in point of real explanatory power, for all the occult entities which heposits under such names as “redness.”

One means by which McX might naturally have tried to impose hisontology of universals on us was already removed before we turned tothe problem of universals. McX cannot argue that predicates such as“red” or “is-red,” which we all concur in using, must be regarded asnames each of a single universal entity in order that they be meaningfulat all. For, we have seen that, being a name of something is a muchmore special feature than being meaningful. He cannot even chargeus—at least not by that argument—with having posited an attribute ofpegasizing by our adoption of the predicate “pegasizes.”

However, McX hits upon a different stratagem. “Let us grant,” hesays, “this distinction between meaning and naming of which you makeso much. Let us even grant that “is red,” “pegasizes,” etc., are not



names of attributes. Still, you admit they have meanings. But thesemeanings, whether they are named or not, are still universals, and Iventure to say that some of them might even be the very things that Icall attributes, or something to much the same purpose in the end.”

For McX, this is an unusually penetrating speech; and the only wayI know to counter it is by refusing to admit meanings. However, I feelno reluctance towards refusing to admit meanings, for I do not therebydeny that words and statements are meaningful. McX and I may agreeto the letter in our classification of linguistic forms into the meaningfuland the meaningless, even though McX construes meaningfulness asthe having (in some sense of “having”) of some abstract entity whichhe calls a meaning, whereas I do not. I remain free to maintain that thefact that a given linguistic utterance is meaningful (or significant, as Iprefer to say so as not to invite hypostasis of meanings as entities) isan ultimate and irreducible matter of fact; or, I may undertake to ana-lyze it in terms directly of what people do in the presence of the lin-guistic utterance in question and other utterances similar to it.

The useful ways in which people ordinarily talk or seem to talk aboutmeanings boils down to two: the having of meanings, which is signifi-cance, and sameness of meaning, or synonomy. What is called givingthe meaning of an utterance is simply the uttering of a synonym,couched, ordinarily, in clearer language than the original. If we are al-lergic to meanings as such, we can speak directly of utterances as sig-nificant or insignificant, and as synonymous or heteronymous one withanother. The problem of explaining these adjectives “significant” and“synonymous” with some degree of clarity and rigor—preferably, as Isee it, in terms of behaviour—is as difficult as it is important. But theexplanatory value of special and irreducible intermediary entities calledmeanings is surely illusory.

Up to now I have argued that we can use singular terms significantlyin sentences without presupposing that there be the entities which those



terms purport to name. I have argued further that we can use generalterms, e.g., predicates, without conceding them to be names of abstractentities. I have argued further that we can view utterances as significant,and as synonymous or heteronymous with one another, without coun-tenancing a realm of entities called meanings. At this point McX beginsto wonder whether there is any limit at all to our ontological immunity.Does nothing we may say commit us to the assumption of universalsor other entities which we may find unwelcome?

I have already suggested a negative answer to this question, in speak-ing of bound variables, or variables of quantification, in connectionwith Russell’s theory of descriptions. We can very easily involve our-selves in ontological commitments by saying, e.g., that there is some-thing (bound variable) which red houses and sunsets have in common;or that there is something which is a prime number between 1000 and1010. But this is, essentially, the only way we can involve ourselves inontological commitments: by our use of bound variables. The use ofalleged names is no criterion, for we can repudiate their namehood atthe drop of a hat unless the assumption of a corresponding entity canbe spotted in the things we affirm in terms of bound variables. Namesare, in fact, altogether immaterial to the ontological issue, for I haveshown, in connection with “Pegasus” and “pegasize,” that names canbe converted to descriptions, and Russell has shown that descriptionscan be eliminated. Whatever we say with help of names can be said ina language which shuns names altogether. To be is, purely and simply,to be the value of a variable. In terms of the categories of traditionalgrammar, this amounts roughly to saying that to be is to be in the rangeof reference of a pronoun. Pronouns are the basic media of reference;nouns might better have been named pro-pronouns. The variables ofquantification, “something,” “nothing,” “everything,” range over ourwhole ontology, whatever it may be; and we are convicted of a particularontological presupposition if, and only if, the alleged presuppositum has



to be reckoned among the entities over which our variables range inorder to render one of our affirmations true.

We may say, e.g., that some dogs are white and not, thereby commitourselves to recognizing either doghood or whiteness as entities. “Somedogs are white” says that some things that are dogs are white; and, inorder that this statement be true, the things over which the bound vari-able “something” ranges must include some white dogs, but need notinclude doghood or whiteness. On the other hand, when we say thatsome zoological species are cross-fertile, we are committing ourselvesto recognizing as entities the several species themselves, abstractthough they be. We remain so committed at least until we devise someway of so paraphrasing the statement as to show that the seeming ref-erence to species on the part of our bound variable was an avoidablemanner of speaking.

If I have been seeming to minimize the degree to which in our philo-sophical and unphilosophical discourse we involve ourselves in onto-logical commitments, let me then emphasize that classical mathematics,as the example of primes between 1000 and 1010 clearly illustrates, isup to its neck in commitments to an ontology of abstract entities. Thusit is that the great mediæval controversy over universals has flared upanew in the modern philosophy of mathematics. The issue is clearernow than of old, because we now have a more explicit standardwhereby to decide what ontology a given theory or form of discourseis committed to: a theory is committed to those and only those entitiesto which the bound variables of the theory must be capable of referringin order that the affirmations made in the theory be true.

Because this standard of ontological presupposition did not emergeclearly in the philosophical tradition, the modern philosophical math-ematicians have not on the whole recognized that they were debatingthe same old problem of universals in a newly clarified form. But the



fundamental cleavages among modern points of view on foundationsof mathematics do come down pretty explicitly to disagreements as tothe range of entities to which the bound variables should be permittedto refer.

The three main mediæval points of view regarding universals aredesignated by historians as realism, conceptualism and nominalism.Essentially these same three doctrines reappear in twentieth-centurysurveys of the philosophy of mathematics under the new names logi-cism, intuitionism and formalism.

Realism, as the word is used in connection with the mediæval con-troversy over universals, is the Platonic doctrine that universals or ab-stract entities have being independently of the mind; the mind maydiscover them but cannot create them. Logicism, represented by suchlatter-day Platonists as Frege, Russell, Whitehead, Church and Carnap,condones the use of bound variables to refer to abstract entities knownand unknown, specifiable and unspecifiable, indiscriminately.

Conceptualism holds that there are universals but they are mind-made. Intuitionism, espoused in modern times in one form or anotherby Poincaré, Brouwer, Weyl and others, countenances the use of boundvariables to refer to abstract entities only when those entities are ca-pable of being cooked up individually from ingredients specified inadvance. As Fraenkel has put it, logicism holds that classes are dis-covered while intuitionism holds that they are invented—a fair state-ment indeed of the old opposition between realism and conceptualism.This opposition is no mere quibble; it makes an essential differencein the amount of classical mathematics to which one is willing to sub-scribe. Logicists, or realists, are able on their assumptions to get Can-tor’s ascending orders of infinity; intuitionists are compelled to stopwith the lowest order of infinity, and, as an indirect consequence, toabandon even some of the classical laws of real numbers. The modern



controversy between logicism and intuitionism arose, in fact, from dis-agreements over infinity.

Formalism, associated with the name of Hilbert, echoes intuitionismin deploring the logicist’s unbridled recourse to universals. But formal-ism also finds intuitionism unsatisfactory. This could happen for eitherof two opposite reasons. The formalist might, like the logicist, objectto the crippling of classical mathmatics; or he might, like the nominal-ists of old, object to admitting abstract entities at all, even in the re-strained sense of mind-made entities. The upshot is the same: theformalist keeps classical mathematics as a play of insignificant nota-tions. This play of notations can still be of utility—whatever utility ithas already shown itself to have as a crutch for physicists and technol-ogists. But utility need not imply significance, in any literal linguisticsense. Nor need the marked success of mathematicians in spinning outtheorems, and in finding objective bases for agreement with one an-other’s results, imply significance. For an adequate basis for agreementamong mathematicians can be found simply in the rules which governthe manipulation of the notations—these syntactical rules being, unlikethe notations themselves, quite significant and intelligible.2

I have argued that the sort of ontology we adopt can be consequen-tial—notably in connection with mathematics, although this is only anexample. Now how are we to adjudicate among rival ontologies? Cer-tainly the answer is not provided by the semantical formula “To be isto be the value of a variable”; this formula serves rather, conversely, intesting the conformity of a given remark or doctrine to a prior ontolog-ical standard. We look to bound variables in connection with ontologynot in order to know what there is, but in order to know what a givenremark or doctrine, ours or someone else’s, says there is; and this muchis quite properly a problem involving language. But what there is is an-other question.



In debating over what there is, there are still reasons for operatingon a semantical plane. One reason is to escape from the predicamentnoted at the beginning of the paper: the predicament of my not beingable to admit that there are things which McX countenances and I donot. So long as I adhere to my ontology, as opposed to McX’s, I cannotallow my bound variables to refer to entities which belong to McX’sontology and not to mine. I can, however, consistently describe ourdisagreement by characterizing the statements which McX affirms.Provided merely that my ontology countenances linguistic forms, orat least concrete inscriptions and utterances, I can talk about McX’ssentences.

Another reason for withdrawing to a semantical plane is to find com-mon ground on which to argue. Disagreement in ontology involvesbasic disagreement in conceptual schemes; yet McX and I, despite thesebasic disagreements, find that our conceptual schemes converge suffi-ciently in their intermediate and upper ramifications to enable us tocommunicate successfully on such topics as politics, weather and, inparticular, language. In so far as our basic controversy over ontologycan be translated upward into a semantical controversy about wordsand what to do with them, the collapse of the controversy into ques-tion-begging may be delayed.

It is no wonder, then, that ontological controversy should tend intocontroversy over language. But we must not jump to the conclusionthat what there is depends on words. Translatability of a question intosemantical terms is no indication that the question is linguistic. To seeNaples is to bear a name which, when prefixed to the words “seeNaples,” yields a true sentence; still there is nothing linguistic aboutseeing Naples.

Our acceptance of an ontology is, I think, similar in principle to ouracceptance of a scientific theory, say, a system of physics: we adopt, at



least insofar as we are reasonable, the simplest conceptual scheme intowhich the disordered fragments of raw experience can be fitted andarranged. Our ontology is determined once we have fixed upon theoverall conceptual scheme which is to accommodate science in thebroadest sense; and the considerations which determine a reasonableconstruction of any part of that conceptual scheme, e.g., the biologicalor the physical part, are not different in kind from the considerationswhich determine a reasonable construction of the whole. To whateverextent the adoption of any system of scientific theory may be said tobe a matter of language, the same—but no more—may be said of theadoption of an ontology.

But simplicity, as a guiding principle in constructing conceptualschemes, is not a clear and unambiguous idea; and it is quite capableof presenting a double or multiple standard. Imagine, e.g., that we havedevised the most economical set of concepts adequate to the play-by-play reporting of immediate experience. The entities under thisscheme—the values of bound variables—are, let us suppose, individualsubjective events of sensation or reflection. We should still find, nodoubt, that a physicalistic conceptual scheme, purporting to talk aboutexternal objects, offers great advantages in simplifying our overall re-ports. By bringing together scattered sense events and treating them asperceptions of one object, we reduce the complexity of our stream ofexperience to a manageable conceptual simplicity. The rule of simplic-ity is indeed our guiding maxim in assigning sense data to objects: weassociate an earlier and a later round sensum with the same so-calledpenny, or with two different so-called pennies, in obedience to the de-mands of maximum simplicity in our total world-picture.

Here we have two competing conceptual schemes, a phenomenalis-tic one and a physicalistic one. Which should prevail? Each has its ad-vantages; each has its special simplicity in its own way. Each, I suggest,



deserves to be developed. Each may be said, indeed, to be the morefundamental, though in different senses: the one is epistemologically,the other physically, fundamental.

The physical conceptual scheme simplifies our account of experiencebecause of the way myriad scattered sense events come to be associatedwith single so-called objects; still there is no likelihood that each sen-tence about physical objects can actually be translated, however devi-ously and complexly, into the phenomenalistic language. Physicalobjects are postulated entities which round out and simplify our accountof the flux of experience, just as the introduction of irrational numberssimplifies laws of arithmetic. From the point of view of the conceptualscheme of the elementary arithmetic of rational numbers alone, thebroader arithmetic of rational and irrational numbers would have the sta-tus of a convenient myth, simpler than the literal truth (namely, the arith-metic of rationals) and yet containing that literal truth as a scattered part.Similarly, from a phenomenalistic point of view, the conceptual schemeof physical objects is a convenient myth, simpler than the literal truthand yet containing that literal truth as a scattered part.

Now what of classes or attributes of physical objects, in turn? Aplatonistic ontology of this sort is, from the point of view of a strictlyphysicalistic conceptual scheme, as much of a myth as that physical-istic conceptual scheme itself was for phenomenalism. This highermyth is a good and useful one, in turn, in so far as it simplifies our ac-count of physics. Since mathematics is an integral part of this highermyth, the utility of this myth for physical science is evident enough. Inspeaking of it nevertheless as a myth, I echo that philosophy of math-ematics to which I alluded earlier under the name of formalism. Butmy present suggestion is that an attitude of formalism may with equaljustice be adopted toward the physical conceptual scheme, in turn, bythe pure aesthete or phenomenalist.



The analogy between the myth of mathematics and the myth ofphysics is, in some additional and, perhaps, fortuitous ways, strikinglyclose. Consider, for example, the crisis which was precipitated in thefoundations of mathematics, at the turn of the century, by the discoveryof Russell’s paradox and other antinomies of set theory. These contra-dictions had to be obviated by unintuitive, ad hoc devices; our math-ematical myth-making became deliberate and evident to all. But whatof physics? An antinomy arose between the undular and the corpus-cular accounts of light; and if this was not as out-and-out a contradic-tion as Russell’s paradox, I suspect that the reason is merely thatphysics is not as out-and-out as mathematics. Again, the second greatmodern crisis in the foundations of mathematics—precipitated in 1931by Godel’s proof that there are bound to be undecidable statements inarithmetic—has its companion-piece in physics in Heisenberg’s inde-terminacy principle.

In earlier pages I undertook to show that some common argumentsin favour of certain ontologies are fallacious. Further, I advanced anexplicit standard whereby to decide what the ontological commitmentsof a theory are. But the question what ontology actually to adopt stillstands open, and the obvious counsel is tolerance and an experimentalspirit. Let us by all means see how much of the physicalistic conceptualscheme can be reduced to a phenomenalistic one; still, physics also nat-urally demands pursuing, irreducible in toto though it be. Let us seehow, or to what degree, natural science may be rendered independentof platonistic mathematics; but let us also pursue mathematics anddelve into its platonistic foundations.

From among the various conceptual schemes best suited to thesevarious pursuits, one—the phenomenalistic—claims epistemologicalpriority. Viewed from within the phenomenalistic conceptual scheme,the ontologies of physical objects and mathematical objects are myths.



The quality of myth, however, is relative; relative, in this case, to theepistemological point of view. This point of view is one among various,corresponding to one among our various interests and purposes.

NOTES

1. This is a revised version of a paper which was presented before theGraduate Philosophy Club of Yale University on May 7, 1948. The latterpaper, in turn, was a revised version of one which was presented before theGraduate Philosophical Seminary of Princeton University on March 15,1951.

2. See Goodman and Quine, “Steps toward a constructive nominalism”Journal of Symbolic Logic, vol. 12 (1947), pp. 97–122.



QUESTIONS

V. LOGIC AND REALITY

1. Provide an example of a counterfactual and indicate how its truth wouldbe determined.

2. Why do we believe the laws of logic?

3. What does Quine mean by his claim that “to be is to be the value of avariable”?

211


ABOUT THE CONTRIBUTORS

1. Lewis Carroll was the pseudonym of Charles Lutwidge Dodgson, whotaught mathematics at Oxford University and wrote the immortal children’sbooks Alice’s Adventures in Wonderland (1865) and its sequel Through theLooking-Glass and What Alice Found There (1872).

2. W. J. Rees was a Senior Lecturer at Leeds University.3. J. F. Thomson was Professor of Philosophy at M.I.T.4. A. N. Prior was a Fellow at Balliol College, Oxford University.5. J. T. Stevenson is Emeritus Professor of Philosophy at the University

of Toronto.6. Nuel D. Belnap is Professor of Philosophy at the University of Pittsburgh.7. Vann McGee is Professor of Philosophy at M.I.T.8. E. J. Lowe is Professor of Philosophy at Durham University.9. D. E. Over teaches at the School of Humanities and Social Sciences at

University of Sunderland.10. Gilbert Ryle was Waynflete Professor of Metaphysical Philosophy at

Oxford University.11. Richard Taylor was Professor of Philosophy at the University of

Rochester.12. Steven M. Cahn, co-editor of this book, is Professor of Philosophy at

the City University of New York Graduate Center.13. Susan Haack is Professor of Law and Philosophy at the University of

Miami.14. Nelson Goodman was Professor of Philosophy at Harvard University.15. Willard V. Quine was Professor of Philosophy at Harvard University.

213


SOURCE CREDITS

Lewis Carroll, “What the Tortoise said to Achilles,” Mind 4 (1895): 278–280.

W. J. Rees, “What Achilles Said to the Tortoise,” Mind 60 (1951): 241–246.J. F. Thomson, “What Achilles Should have Said to the Tortoise,” Ratio 3

(1960): 95–105.A. N. Prior, “The Runabout Inference Ticket,” Analysis 21 (1960): 38–39.J. T. Stevenson, “Roundabout the Runabout Inference Ticket,” Analysis 21

(1961): 124–128.Nuel D. Belnap, “Tonk, Plonk, and Plink,” Analysis 22 (1962): 130–134.Vann McGee, “A Counterexample to Modus Ponens,” Journal of Philoso-

phy 82 (1985): 462–471.E. J. Lowe, “Not A Counterexample to Modus Ponens,” Analysis 47

(1987): 44–47.D. E. Over, “Assumptions and the Supposed Counterexamples to Modus

Ponens,” Analysis 47 (1987) 142–146.Gilbert Ryle, “It Was to Be,” from Gilbert Ryle, Dilemmas (Cambridge

University Press, 1954), pp. 15–35.Richard Taylor, “Fatalism,” The Philosophical Review 71 (1962): 56–66.Steven M. Cahn and Richard Taylor, “Time, Truth, and Ability,” Analysis

25 (1964): 137–141.Susan Haack, “The Justification of Deduction,” Mind 85 (1976): 112–119.Nelson Goodman, “The Problem of Counterfactual Conditionals,” Journal

of Philosophy 44 (1947): 113–120.Willard V. Quine, “On What There Is,” Review of Metaphysics 2 (1948):

21–38.

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